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Statistical modelling for a new family of generalized distributions with real data applications


  • The modern trend in distribution theory is to propose hybrid generators and generalized families using existing algebraic generators along with some trigonometric functions to offer unique, more flexible, more efficient, and highly productive G-distributions to deal with new data sets emerging in different fields of applied research. This article aims to originate an odd sine generator of distributions and construct a new G-family called "The Odd Lomax Trigonometric Generalized Family of Distributions". The new densities, useful functions, and significant characteristics are thoroughly determined. Several specific models are also presented, along with graphical analysis and detailed description. A new distribution, "The Lomax cosecant Weibull" (LocscW), is studied in detail. The versatility, robustness, and competency of the LocscW model are confirmed by applications on hydrological and survival data sets. The skewness and kurtosis present in this model are explained using modern graphical methods, while the estimation and statistical inference are explored using many estimation approaches.

    Citation: M. E. Bakr, Abdulhakim A. Al-Babtain, Zafar Mahmood, R. A. Aldallal, Saima Khan Khosa, M. M. Abd El-Raouf, Eslam Hussam, Ahmed M. Gemeay. Statistical modelling for a new family of generalized distributions with real data applications[J]. Mathematical Biosciences and Engineering, 2022, 19(9): 8705-8740. doi: 10.3934/mbe.2022404

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  • The modern trend in distribution theory is to propose hybrid generators and generalized families using existing algebraic generators along with some trigonometric functions to offer unique, more flexible, more efficient, and highly productive G-distributions to deal with new data sets emerging in different fields of applied research. This article aims to originate an odd sine generator of distributions and construct a new G-family called "The Odd Lomax Trigonometric Generalized Family of Distributions". The new densities, useful functions, and significant characteristics are thoroughly determined. Several specific models are also presented, along with graphical analysis and detailed description. A new distribution, "The Lomax cosecant Weibull" (LocscW), is studied in detail. The versatility, robustness, and competency of the LocscW model are confirmed by applications on hydrological and survival data sets. The skewness and kurtosis present in this model are explained using modern graphical methods, while the estimation and statistical inference are explored using many estimation approaches.



    Since the last few decades, almost all research about the origination of G-families is just adopting the approaches of differential equations, compounding, weighting, etc., and thousands of statistical models have been added to the literature. No doubt, some very useful models are introduced using the above-described techniques. Still, a keen analysis reveals that mostly out of these models are internally correlated and maybe the replacement of one another in definite parametric conditions. Moreover, they are similar in mathematical appearance with only mild differences. The critical point is that almost all models are algebraic and non-trigonometric. For a brief study, we refer the reader to [1,2,3,4,5]. Also, for more information about machine learning, see [6,7,8].

    Recently, the attention of statisticians has turned towards the directional data and disbursing the trigonometric functions in the existing classical models in order to construct generalized trigonometric families of distributions which open new and co-research horizons for Mathematicians and Statisticians. It is observed that the trigonometric functions enhance the flexibility prominently, keep relative balance and simplicity, show vast applicability in modeling different types of practical data sets and explore the skewness, kurtosis, and tail characteristics along with improving the goodness-of-fit (GoF). Table 1 presents chronological literature review on sine function based families and distributions.

    Table 1.  Sine generalized distributions and families in chronological order.
    S.No. Introducer(s)(year) Sine based distributions and G-families.
    1 Chakraborty et al. (2012) explored the sin-skew logistic distribution.
    2 Souza (2015) suggested new trigonometric classes of probabilistic distributions.
    3 Kharazmi and Saadatinik (2016) presented hyperbolic sine-Weibull distribution.
    4 Chesneau et al. (2018) explored the cosine-sine distribution.
    5 Mahmood et al. (2019) presented a new sine-G family of distributions.
    6 Chesneau et al. (2020) deduced sine kumaraswamy-G family of distributions.
    7 Al-Babtain et al. (2020) introduced sine topp-leone-G family of distributions.
    8 Nagarjuna et al. (2021) worked on sine power lomax model.
    9 Shrahili et al. (2021) did the estimation of sine inverse exponential model.
    10 Shrahili et al. (2021) introduced sine half-logistic inverse rayleigh distribution.
    11 Gang Shi et al. (2021) presented sine entropy of uncertain random variables.

     | Show Table
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    The development of trigonometric and algebraic functions mixed with a new generalized Lomax family of probability distributions is the basic motivation. The remainder motivations are five folded:

    ● To develop a new sine generator of distributions using the spirit of odd generator and combination of algebraic and trigonometric functions concurrently;

    ● To introduce a new G-family called "The Odd Lomax Trigonometric Generalized Family of Distributions" (Locsc-G for short) in a trigonometric scenario;

    ● The proposed family is simple, free from non-identifiability and over parametrization issues;

    ● To investigate the injection of sine-cosecant functions in odd generator methods in classical distributions, leading to a novel, more versatile, and effective models;

    ● The new density adopts uni modal features or shapes as well (in almost all base models), and the hazard function adopts all monotone and non-monotone shapes.

    The current study is conducted following the spirit of the odd generator presented by [9], the Weibull-G family developed by [10], the sine-G family introduced by [11] and the generalized odd Gamma-G family introduced by [12] collectively. In modern distribution theory, in our view, the trigonometric functions based on generalized families and distributions will prove a breakthrough for modeling the data of physical phenomena.

    In Section 1, an introduction about trigonometric work with motivations are presented. New generator and family with special members are presented in Section 2, whereas the new family characteristics are derived in Section 3. In Section 5, the graphical behavior of the new family is observed using famous statistical models. The special member using Weibull as the baseline is investigated in Section 6 along with sub-models. Two data applications demonstrate the significance of the new family and model in Section 8 while final remarks and conclusions end the study in Section 9.

    The odd generator (G(x)1G(x)) and sine function [sin(π2G(x))] are used collectively to develop the new generator.

    W[G(x)]=sin(π2G(x))1sin(π2G(x))=(1sin(π2G(x))sin(π2G(x)))1=(csc(π2G(x))1)1 (2.1)

    This generator W[G(x)]:[0,1]R (a link function) satisfies all required conditions of T-X family of distributions.

    Let r(t)=αλ[1+(tλ)](α+1) is the lomax density where 0<t<. Replace "t" by the new generator W[G(x)]=[csc(π2G(x))1]1 in lomax function, we arrived at the new family "Locsc-G" whose distribution function in Eq (2.2), and probability density function in Eq (2.3) and hazard rate function in Eq (2.4), respectively, are given below.

    F(x)=[csc(π2G(x))1]10r(t)dt=1{1+λ1[csc(π2G(x))1]1}α (2.2)
    f(x)=απg(x)csc(π2G(x))cot(π2G(x))2λ[csc(π2G(x))1]2{1+λ1[csc(π2G(x))1]1}(α+1) (2.3)
    h(x)=απg(x)csc(π2G(x))cot(π2G(x))2λ[csc(π2G(x))1]2{1+λ1[csc(π2G(x))1]1}1 (2.4)

    In Table 2, for example, eight new recruits are added by employing the well-known statistical distributions on all feasible intervals.

    Table 2.  Specific models of the new family.
    cdf G(x) Support New cdf F(x) Parameters
    Uniform (0,θ) 1[1+(λ1(csc(π2(xθ))1)1)]α (λ,α,θ)
    Exponential (0,) 1[1+(λ1(csc(π2(1eβx))1)1)]α (λ,α,β)
    Weibull (0,) 1[1+(λ1(csc(π2(1e(βx)σ))1)1)]α (λ,α,β,σ)
    Frechet (0,) 1[1+(λ1(csc(π2(e(βx)σ))1)1)]α (λ,α,β,σ)
    Burr XII (0,) 1[1+(λ1(csc(π2(1[1+(x/s)c]k))1)1)]α (λ,α,c,k,s)
    Logistic R 1[1+(λ1(csc(π2([1+e(xμ)/s]1))1)1)]α (λ,α,μ,s)
    Gumbel R 1[1+(λ1(csc(π2(ee(xμ)/σ))1)1)]α (λ,α,μ,σ)
    Normal R 1[1+(λ1(csc(π2Φ((xμ)/σ))1)1)]α (λ,α,μ,σ)

     | Show Table
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    We will discuss a characteristic of X called the quantile function (qf), which may be determined directly by inverting (2.2) as below

    Q(u)=G1{2πsin1[(λ(1u)1α1)1+1]1} (3.1)

    Equation (3.1) possesses a lot of applications; some are given below: 1) To find median, quartiles, deciles and percentiles. 2) Replacing any standard model, this equation can be used to simulate density, histogram, and exact cdfs for these data can be accomplished. 3) The variability analysis related to skewness and kurtosis can be performed on the basis of quantile measures as the Bowley skewness (see [13]) and the Moors kurtosis (see [14]), respectively. A remarkable function related to Eq (3.1), having statistical significance discussed in [15], is the quantile density function denoted by Q(U) is:

    Q(U)=2λ(1u)(1+αα)[λ(1u)(1α)1]2πα[(λ(1u)(1α)1)1+1]2[1((λ(1u)(1α)1)1+1)2]12

    The hazard rate function, which is an essential concept that plays a vital role in risk and survival analysis, is an example of an important function. There are some other important functions such as survival function S(x), also another important one is reversed hazard rate r(x), at last we must not forget the cumulative hazard rate H(x), and the very interesting mills' ratio m(x), elasticity e(x) and finally the conditional reliability function ˉG(G(x),α,β|t), which are respectively, presented below.

    S(x)={1+λ1[csc(π2G(x))1]1}α
    h(x)=απg(x)csc(π2G(x))cot(π2G(x))2λ[csc(π2G(x))1]2{1+λ1[csc(π2G(x))1]1}1
    r(x)=απg(x)csc(π2G(x))cot(π2G(x))2λ[csc(π2G(x))1]2{1+λ1[csc(π2G(x))1]1}(α+1)1{1+λ1[csc(π2G(x))1]1}α
    m(x)=2λ[1+λ1(csc(π2G(x))1)1]απg(x)csc(π2G(x))cot(π2G(x))(csc(π2G(x))1)2
    H(x)=log[{1+λ1[csc(π2G(x))1]1}α]1
    e(x)=(ln(G(x)))ln{1[1+(λ1(csc(π2G(x))1)1)]α}
    ˉG(G(x),α,λ|t)=ˉG(G(x+t),α,λ)ˉG(G(t),α,λ)=[1+λ1(csc(π2G(t))1)1]α[1+λ1(csc(π2G(x+t))1)1]α

    We have the following linear representations for the new families CDF and pdf.

    Proposition 3.1. The new family's cdf and pdf have the following linear representations:

    F(x)=k=0wi,jG(x)2k,f(x)=k=0wi,j[2kg(x)G(x)2k1], (3.2)

    where

    wi,j=i,j=0(π2)2k(1)i+j+1(λ)i(αi)(ij)ak(j) (3.3)

    Proof. If the cdf and pdf of a random variable Y can be stated as Hc(x)=G(x)c and hc(x)=cG(x)c1g(x) then we say that this random variable has exp-G with power parameter c>0, The cdf of the new family, required to be linearized, is

    F(x)=1{1+λ1[csc(π2G(x))1]1}α

    using the expansion (1+x)n=i=0(ni)xi and binomial expansion simultaneously, F(x) becomes

    F(x)=i=1j=0(1)i+j+1(λ)i(αi)(ij)[csc(π2G(x))]j

    using MATHEMATICA 11.1, [csc(π2G(x))]j=k=0ak(j)(π2G(x))2k

    where a0(j)=1, a1(j)=j/6, a2(j)=(j/180)+(j2/72), etc.

    For F(x), the required linear representation is obtained. Moreover, just by simple differentiation, the linear representation of f(x) can be obtained.

    For the new family, the rth moment is (r be an integer and all sum and integrals are assumed to exist)

    μr=xrf(x)dx=xrk=0wi,j(2k)g(x)G(x)2k1dx=k=0wi,jE(Xrk) (3.4)

    μr is also expressed by consuming the quantile function (or changing the variable x=Q(p)) given by Eq (3.1), in this way

    μr=10[Q(p)]rdp.=10{QG[2πsin1((λ(1u)(1α)1)1+1)1]}. (3.5)

    The derived integral is computable using any modern mathematical software like Mathematica, R, Matlab, or Maple for given G(x), α, and λ.

    For r1,s0, routinely, the (r,s)th probability weighted moment (PWM) is expressed as

    ρr,s=E[XrF(X)s]=0xrF(x)sf(x)dx. (3.6)

    Then, we have

    F(x)s=[1[1+λ1(csc(π2G(x))1)1]α]s. (3.7)

    After a bit modification using trigonometric relations, (3.7) can be written as:

    F(x)s=[1[1+λ1(sin(π2G(x))1sin(π2G(x)))]α]s

    and

    f(x)=απg(x)csc(π2G(x))cot(π2G(x))2λ(csc(π2G(x))1)2[1+λ1(csc(π2G(x))1)1](α+1).

    Similarly,

    f(x)=απg(x)cos(π2G(x))2λ(1sin(π2G(x)))2[1+λ1(sin(π2G(x))(1sin(π2G(x))))](α+1)

    in Eq (3.6), ρr,s can be written as:

    ρr,s=l,m=0w(i,j,k)xr(2l+m+1)g(x)(G(x))(2l+m+1)1dx. (3.8)
    ρr,s=l,m=0w(i,j,k)xrh(2l+m+1)(x)dx. (3.9)
    ρr,s=l,m=0w(i,j,k)E(Xr(l,m)) (3.10)

    where

    w(i,j,k)=i,j,k=0α(2l)!(2l+m+1)(π2)(2l+m+1)(1)(i+k+l)(λ)(α(i+1)2)(si)((α(i+1)1)j)((α(i+1)1)k)Cm(α(i+1)+k1).

    Introducing mathematical properties is very important. The moment generating function is stated in the following mathematical format:

    M(t)=+etxf(x)dx=+r=0trr!μr.

    Otherwise, without utilizing the moments but consuming the linear representation presented in expression (3.2), M(t) is expressed as

    M(t)=+k=0wi,j+etx2kg(x)G(x)2k1dx=+k=0wi,j(2k)10p(2k1)etQG(p)dp.

    By solving the following equation log[f(x)]x=0 and log[h(x)]x=0 we can provide the density and hazard rate function critical points respectively. For the Locsc-G density function, the nonlinear equation related to density is:

    g(x)g(x)π(α+1)g(x)cot(12πG(x))csc(12πG(x))(csc(12πG(x))1)3(1(csc(12πG(x))1)2+λ)12πg(x)cot(12πG(x))πg(x)csc(πG(x))+πg(x)cot(12πG(x))csc(12πG(x))csc(12πG(x))1=0. (3.11)

    While the critical points of the hrf are obtained from the equation log[h(x)]x=0

    g(x)g(x)+14πg(x)(sin(πG(x))csc3(12πG(x))(2λcsc(12πG(x))2λ+3)(csc(12πG(x))1)(λcsc(12πG(x))))(14πg(x)((λ+1)2tan(12πG(x))4csc(πG(x))))=0. (3.12)

    A detailed description of stochastic ordering is available in [16], here, utilizing the family parameters α and λ, a proof is presented concerning the stochastic ordering.

    Proposition 3.2. let us suppose that we heave a random variable let us say it X came from a distribution with the density function f1(x) as defined in (2.3) with parameters α1 and λ and let us suppose that we heave a random variable let us say it Y came from a distribution with the density function as defined in f2(x) as defined in (2.3) with parameters α2 and λ.So, if α1α2, we have XlrY, i.e., f1(x)f2(x) is decreasing.

    Proof. The density is

    f(x)=απg(x)csc(π2G(x))cot(π2G(x))2λ(csc(π2G(x))1)2[1+λ1(csc(π2G(x))1)1](α+1).

    Then

    f1(x)f2(x)=(α1α2)[1+λ1(csc(π2G(x))1)1](α1α2).

    Since α1α2, then after differentiation with respect to x, we get

    xf1(x)f2(x)=(α1α2)(α1α2)π2λg(x)csc(π2G(x))cot(π2G(x))(csc(π2G(x))1)2[1+λ1(csc(π2G(x))1)1](α1α2)10.

    The proof of Proposition 3.2 ends with the conclusion that XlrY.

    The reliability parameter is very important. See [17] for a detailed study on stress-strength reliability.

    Let us suppose that we heave a random variable X came from a distribution with density function f1(x) given by (2.3) with parameters α1 and λ1 and another variable Y came from a distribution with distribution function F2(x) defined as (2.2) with parameters α2 and λ2. Then, the reliability parameter is defined by :

    R=P(Y<X)=f1(x)F2(x)dx.

    With f1(x) and F2(x) functions,

    R=α1πg(x)cos(π2G(x))2λ1(1sin(π2G(x)))2[1+λ11(sin(π2G(x))1sin(π2G(x)))](α+1)(1[1+λ21(sin(π2G(x))1sin(π2G(x)))]α2)dx.

    After simplification, we get

    R=k,l,n=0V(i,j,m)g(x)G(x)2(k+l+n)1dx.

    Where

    V(i,j,m)=i=1j=0m=0(1)(k+m+1)(α1π(λ1)(j1)(λ2)i(2k)!2)(α2i)((α2+1)j)((i+j+2)m)(π2)2(k+l+n)dl(i+j)en(m)

    where d0(i+j)=1, d1(i+j)=l/6, d2(i+j)=(l/180)+((l2)/72), etc. and similar for en(m). If α1=α2 and λ1=λ2 (corresponds to the case being distributed identically), at end, we obtained R=12(k+l+n).

    Order statistics invariably appear in a variety of applications requiring data related to survival testing. You will find all the information in the book [18].

    Consider the ith order statistic Xi:n and its density is to find. Let a random sample X1,,Xn is chosen from the new family then

    fi:n(x)=n!(i1)!(ni)!f(x)F(x)i1[1F(x)]ni,xR. (3.13)

    Equations (2.2) and (2.3) are substituted in the Eq (3.13), we get

    fi:n(x)=n!(i1)!(ni)!απg(x)cos(π2G(x))2λ(1sin(π2G(x)))2[1+λ1(sin(π2G(x))1sin(π2G(x)))](α+1){1[1+λ1(sin(π2G(x))1sin(π2G(x)))]α}i1{[1+λ1(sin(π2G(x))1sin(π2G(x)))]α}ni.

    Notably, f1:n(x) and fn:n(x) are the densities of X1:n=inf(X1,,Xn) and Xn:n=sup(X1,,Xn) respectively.

    Proposition 3.3. TheXi:n pdf may be represented as a linear combination of pdfs from the exp-G distribution family.

    Proof. Firstly, consider the Eq (3.13) which displays the expression of fi:n(x). Applying the binomial series expansion and substituting the Eq (3.2) in Eq (3.13), we get

    fi:n(x)=n!(i1)!(ni)!nij=0(1)j(nij)f(x)F(x)j+i1=n!(i1)!(ni)!nij=0(1)j(nij)απg(x)cos(π2G(x))2λ(1sin(π2G(x)))2[1+λ1(sin(π2G(x))1sin(π2G(x)))](α+1){1[1+λ1(sin(π2G(x))1sin(π2G(x)))]α}j+i1.

    By virtue of generalized binomial expansion and relevant series on sine and cosine trigonometric functions, we have

    fi:n(x)=p,q=0V(j,l,m,s)[2(p+q)+1]g(x)(G(x))[2(p+q)+1]1, (3.14)
    fi:n(x)=p,q=0V(j,l,m,s)h(2(p+q)+1)(x), (3.15)

    where

    V(j,l,m,n)=nij=0+l,m,s=0n!(i1)!(ni)!(1)j+l+p+s(2p)!(2(p+q)+1)α(π2)(2(p+q)+1)(λ)(m1)dq(m+n)(nij)(j+i1l)(α(l+1)1m)((m+2)s)

    Moreover, h(2(p+q)+1)(x) is a pdf of the exp-G family of distributions with parameter (2(p+q)+1), the proposal evidence (3.3) is accomplished.

    In this section, we introduce different classical estimation methods for estimating the new family parameters α, λ, and ξ, which are obtained by maximization of minimization of the objective function, as we will see in this section. For more information about the introduced estimation methods, see [19,20,21].

    The estimated parameters of our proposed family by the maximum likelihood estimation (MLE) method are obtained by maximizing the log-likelihood function of (2.3) which is defined in the following equation.

    (α,λ,ξ)=n[log(πα)(α+1)log(1λ(csc(12πG(xi,ξ))1)2+1)log(2λ)]+ni=1log[g(xi,ξ)]+ni=1log[cot(12πG(xi,ξ))]2ni=1log[csc(12πG(xi,ξ))1]+ni=1log[csc(12πG(xi,ξ))].

    The estimated parameters of our proposed family by Anderson-Darling estimation (ADE) method is obtained by minimizing the following equation (x(1)x(2)x(n))

    A(α,λ,ξ)=n1nni=1(2i1)[logF(xi)+logS(xi)]=n1nni=1(2i1)×[log(1{1+λ1[csc(π2G(xi,ξ))1]1}α)+log{1+λ1[csc(π2G(xi,ξ))1]1}α].

    The estimated parameters of our proposed family by right-tail Anderson-Darling estimation (RADE) method is obtained by minimizing the following equation (x(1)x(2)x(n))

    R(α,λ,ξ)=n22ni=1F(xi:n|α,λ,ξ)1nni=1(2i1)logS(xn+1i:n|α,λ,ξ)=n22ni=1(1{1+λ1[csc(π2G(xi,ξ))1]1}α)1nni=1(2i1)log{1+λ1[csc(π2G(xn+1i,ξ))1]1}α.

    The estimated parameters of our proposed family by Cramér-von Mises estimation (CVME) method is obtained by minimizing the following equation (x(1)x(2)x(n))

    C(α,λ,ξ)=112n+ni=1[F(xi|α,λ,ξ)2i12n]2=112n+ni=1[1{1+λ1[csc(π2G(xi,ξ))1]1}α2i12n]2.

    The estimated parameters of our proposed family by least-squares estimation (LSE) method is obtained by minimizing the following equation (x(1)x(2)x(n))

    V(α,λ,ξ)=ni=1[F(xi|α,λ,ξ)in+1]2=ni=1[1{1+λ1[csc(π2G(xi,ξ))1]1}αin+1]2.

    The estimated parameters of our proposed family by weighted least-squares estimation (WLSE) method is obtained by minimizing the following equation (x(1)x(2)x(n))

    W(α,λ,ξ)=ni=1(n+1)2(n+2)i(ni+1)[F(xi|α,λ,ξ)in+1]2=ni=1(n+1)2(n+2)i(ni+1)[1{1+λ1[csc(π2G(xi,ξ))1]1}αin+1]2.

    The estimated parameters of our proposed family by maximum product of spacing estimation (MPSE) method is obtained by maximizing the following equation (x(1)x(2)x(n))

    H(α,λ,ξ)=1n+1n+1i=1logDi(α,λ,ξ),

    where

    Di(α,λ,ξ)=F(x(i)|α,λ,ξ)F(x(i1)|α,λ,ξ)={1+λ1[csc(π2G(xi1,ξ))1]1}α+{1+λ1[csc(π2G(xi,ξ))1]1}α.

    We presented a few special models of the new family using well-known statistical distributions as a baseline, developed main functions, and analyzed and described graphical flexibility.

    Let X be an exponentiated exponential random variable with cdf G(x)=(1eδx)β and density g(x)=δβeδx(1eδx)(β1). Then the CDF, pdf, and hazard rate function of the LocscEE distribution, respectively, become as (for x>0)

    F(x)=1[1+λ1(csc(π2((1eδx)β))1)1]α
    f(x)=απ(δβeδx(1eδx)(β1))csc(π2(1eδx)β)cot(π2(1eδx)β)2λ(csc(π2(1eδx)β)1)2[1+λ1(csc(π2(1eδx)β)1)1](α+1)
    h(x)=απ(δβeδx(1eδx)(β1))csc(π2(1eδx)β)cot(π2(1eδx)β)2λ(csc(π2(1eδx)β)1)2[1+λ1(csc(π2(1eδx)β)1)1].

    Figure 1 displays some plots of the density and hazard rate function of the LocscEE distribution for some parametric values. Figure 1(a) depicts that the LocscEE density exhibits reverse-j, approximately symmetrical, left-skewed and right-skewed shapes. Figure 1(b) reveals that the LocscEE hazard rate function has increasing, decreasing, increasing-decreasing-increasing, and upside-down bathtub shapes.

    Figure 1.  Plots of the LocscEE (a) density (b) hazard rate for some parametric values.

    Taking G(x) to be the Weibull cdf with scale parameter σ>0 and shape parameter β>0, say G(x)=1eσxβ, and the weibull density g(x)=σβxβ1eσxβ, it follows the four-parameters LocscW having the following new cdf, pdf and hazard rate function (for x>0)

    F(x)=1[1+λ1(csc(π2(1eσxβ))1)1]α
    f(x)=απ(σβxβ1eσxβ)csc(π2(1eσxβ))cot(π2(1eσxβ))2λ(csc(π2(1eσxβ))1)2[1+λ1(csc(π2(1eσxβ))1)1](α+1)
    h(x)=απ(σβxβ1eσ(x)β)csc(π2(1eσxβ))cot(π2(1eσxβ))2λ(csc(π2(1eσxβ))1)2[1+λ1(csc(π2(1eσxβ))1)1].

    Figure 2 displays some plots of the density and hazard rate function of the LocscW distribution for some parametric values. Figure 2(a) depicts that the LocscW density have symmetrical, right-skewed, left-skewed, reversed-J and J shapes. Figure 2(b) reveals that the LocscW hazard rate function has decreased, increasing bathtub and upside-down bathtub shapes.

    Figure 2.  Plots of the LocscW (a) density (b) hazard rate for some parametric values.

    Let X be an burr random variable with pdf g(x)=δβ(x)δ1(1+(x)δ)(β+1) and cdf G(x)=1(1+xδ)β,x>0δ,β>0. it follows the four-parameters LocscB having the following cdf, pdf and hazard function (for x>0)

    F(x)=1[1+λ1(csc(π2(1(1+xδ)β))1)1]α
    f(x)=απδβxδ1(1+xδ)(β+1)csc(π2(1(1+xδ)β))cot(π2(1(1+xδ)β))2λ(csc(π2(1(1+xδ)β))1)2[1+λ1(csc(π2(1(1+xδ)β))1)1](α+1)
    h(x)=απδβ(x)δ1(1+(x)δ)(β+1)csc(π2(1(1+xδ)β))cot(π2(1(1+xδ)β))2λ(csc(π2(1(1+xδ)β))1)2[1+λ1(csc(π2(1(1+xδ)β))1)1)].

    Figure 3 displays some plots of the density and hazard rate function of the LocscB distribution for some parametric values. Figure 3(a) depicts that the LocscB density have symmetrical, right-skewed, left-skewed and reversed-J shapes. Figure 3(b) reveals that the LocscB hazard rate function have decreasing, increasing and upside down bathtub shapes.

    Figure 3.  Plots of the LocscB (a) density (b) hazard rate for some parametric values.

    In this part, we will look at the unique member of the Locsc-G family of distributions that uses the Weibull distribution as a baseline, as well as its key features. As a result, by swapping the CDF G(x)=1eσxβ, x>0, into Eq (2.2), The new distribution's CDF can be written as below

    F(x)=1[1+λ1(csc(π2(1eσxβ))1)1]αx>0. (6.1)

    The corresponding pdf is

    f(x)=απ(σβxβ1eσxβ)csc(π2(1eσxβ))cot(π2(1eσxβ))2λ(csc(π2(1eσxβ))1)2[1+λ1(csc(π2(1eσxβ))1)1](α+1). (6.2)

    The corresponding hazard rate function is

    h(x)=απ(σβxβ1eσxβ)csc(π2(1eσxβ))cot(π2(1eσxβ))2λ(csc(π2(1eσxβ))1)2[1+λ1(csc(π2(1eσxβ))1)1]. (6.3)

    Figure 4 illustrates the suggested model's showing density forms.

    Figure 4.  {Different plots of the LocscW density.}.

    Figure 5 illustrates the suggested model's displaying hazard rate function forms.

    Figure 5.  {Different plots of the LocscW hazard rate function.}.

    The hazard rate function is a key notion and performs a central role in risk and survival analysis. There are some other important functions such survival function S(x), also another important one is reversed hazard rate r(x), at last we must not forget the cumulative hazard rate H(x), and the very interesting mills' ratio m(x), elasticity e(x) and finally the conditional reliability function ˉG(G(x),α,β|t), which are respectively, presented below.

    S(x)=[1+λ1(csc(π2(1eσxβ))1)1]α
    h(x)=απσβxβ1eσxβcsc(π2(1eσxβ))cot(π2(1eσxβ))2λ(csc(π2(1eσxβ))1)2[1+λ1(csc(π2(1eσxβ))1)1]
    r(x)=απσβxβ1eσxβcsc(π2(1eσxβ))cot(π2(1eσxβ))(1[1+λ1(csc(π2(1eσxβ))1)1]α)[1+λ1(csc(π2(1eσxβ))1)1](α+1)2λ(csc(π2(1eσxβ))1)2
    m(x)=2λ[1+λ1(csc(π2(1eσxβ))1)1](csc(π2(1eσxβ))1)2απg(x)csc(π2(1eσxβ))cot(π2(1eσxβ))
    H(x)=log[1F(x)]=log{[1+λ1(csc(π2(1eσxβ))1)1]α}1
    e(x)=(ln(1eσxβ))ln{1[1+λ1(csc(π2(1eσxβ))1)1]α}
    ˉG(G(x),α,λ|t)=[1+λ1(csc(π2(1eσtβ))1)1]α[1+λ1(csc(π2(1eσ(x+t)β))1)1]α.

    The residual life has several uses in probability and statistics and risk assessment. The residual lifetime of LocscW random variable X denoted by Rt(x) is

    Rt(x)=[1+λ1(csc(π2(1eσxβ))1)1]α[1+λ1(csc(π2(1eσ(x+t)β))1)1]α

    Additionally, the reversed hazard rate function ˉRt(x) is

    ˉRt(x)=[1+λ1(csc(π2(1eσxβ))1)1]α[1+λ1(csc(π2(1eσ(xt)β))1)1]α.

    The quantile function of LocscW is given by

    QLocscW(p)=(log(π2sec1(1λ(1p)1/αλ+1))σ)1/β,p(0,1). (6.4)

    The quartiles and octiles, as well as skewness and kurtosis, can be calculated from this description, and the following distribution results are useful: For a random variable U with a uniform distribution on (0,1), QLocscW(U) has the LocscW distribution.

    Furthermore, the quantile density function (qdf) for LocscW may indeed be calculated by getting the differentiation of QLocscW(U) with respect to p. The median, in an instance, is provided as

    Median=QF(0.5)=QG(x)=[1σlog(12πcsc1(1+λ(0.5)1α1))](1β).

    Obtaining skewness is very important for researchers, however MacGillivary (1986) created a technique to obtain it by the aid of the quantile function, such as below

    δ(p)=δ(1)(α,β,σ,λ)δ(2)(α,β,σ,λ)=Q(1p)+Q(p)2Q(1/2)Q(1p)Q(p) (6.5)

    where pϵ(0,1) and Q(.) is the qf stated in Eq (6.4).

    δ(1)(α,β,σ,λ)=[1σlog(12πcsc1(1+λ(p1α1)))](1β)+[1σlog(12πcsc1(1+λ(1p)1α1))](1β)2[1σlog(12πcsc1(1+λ(1/2)1α1))](1β)
    δ(2)(α,β,σ,λ)=[1σlog(12πcsc1(1+λ(p1α1)))](1β)[1σlog(12πcsc1(1+λ(1p)1α1))](1β)

    Because the MacGillivary skewness measure δ(p) is simply dependent on qf, it can efficiently characterize the influence of the parameters (α,β,σ,λ) just on the skewness of X. In Figure 6, the plots in Figure 6(left) describes keeping parameters (α=1,λ=0.1,σ=0.5) as constant while the parameter β values are increased from 0.1 to 0.9, then δ(p)0 means skewness approaches to zero (or approaching to symmetry).

    Figure 6.  MacGillivary's skewness plots for selected values of the parameters.

    In Figure 6, the plots in Figure 6(middle) describes keeping parameters (α=1.5,λ=0.5,σ=1.0) as constant (as compared to Figure 6(left), the values of (α,λ,σ) are increased by 0.5) while the parameter β values are also increased from 0.1 to 1.0 on regular spacing, then δ(p)0.5 means lightly skewness is observed.

    In Figure 6, the plots in Figure 6(right) describes keeping parameters (α=1.5,λ=1.0,σ=1.0) as constant (as compared to Figure 6(middle), the values of (α,σ) are not changes but λ in increased 0.5 only) while the parameter β values are increased from 0.05 to 0.5 on different spacing values, then δ(p)1.0 means significant skewness is produced.

    In Figure 7, the plots in Figure 7(left) describes keeping parameters (α=2.0,β=1.5,σ=0.1) as constant while the parameter λ values are increased from 0.1 to 0.95, then the symmetry is loosed towards left (negative skewness is observed).

    Figure 7.  MacGillivary's skewness plots for selected values of the parameters.

    In Figure 7, the plots in Figure 7(middle) describes keeping parameters (β=1.0,λ=0.1,σ=0.5) as constant while the parameter α values are increased from 0.1 to 1.05 on different spacing values, then δ(p) increases heavily means the highly skewness is produced on right side.

    In Figure 7, the plots in Figure 7(right) describes keeping parameters (α=1.5,λ=0.5,σ=0.5) as constant while the parameter β values are increased from 1.05 to 3.5 on different spacing values, then δ(p) increases means the right skewness is produced.

    Recently, the tendency has shifted, and the graphical image is now more common and preferred than numerical and tabular representation. The 3D figures showed below vividly demonstrate the shift in skewness and kurtosis that occurs when the parental model parameters are changed. In Figure 8, the alternate curves 8(a) and 8(c) are for skewness while 8(b) and 8(d) are for kurtosis respectively. Both measures of skewness and kurtosis for the proposed model are highly dependent on the fixed values of α and λ.

    Figure 8.  For chosen parametric values, curves for skewness and kurtosis.

    In Figure 9, the curves 9(a) and 9(c) are for skewness while 9(b) and 9(d) are for kurtosis respectively.

    Figure 9.  Curves for the skewness and the kurtosis for selected parametric values.

    The baseline parameters σ=2.5 and β=3.1 are taken in Figure 9, it is observed that the skewness is decreased (symmetry is increased) in Figure 9(a) as well as the kurtosis is reduced (normality is increased) in Figure 9(b).

    In Table 3, three new reduced/sub-models of LocscW distribution are deduced here, just limiting the parametric values.

    Table 3.  Reduced models of LocscW distribution.
    S.No. σ β Reduced models of LocscW distribution Comments
    1 1 - Lomax cosecant 1 parameter Weibull (LocscW(1P)) distribution New
    2 - 1 Lomax cosecant exponential (LocscE) distribution New
    3 - 2 Lomax cosecant rayleigh (LocscR) distribution New

     | Show Table
    DownLoad: CSV
    Table 4.  Simulation values of BIAS, MSE and MRE for (σ=0.25, β=0.5, α=0.75, λ=0.5).
    n Est. Est. Par. MLE ADE CVME MPSE LSE RTADE WLSE
    50 BIAS ˆσ 0.166439{7} 0.15964{3} 0.156342{1} 0.156488{2} 0.165037{6} 0.161629{4} 0.16181{5}
    ˆβ 0.203115{3} 0.201461{2} 0.302332{7} 0.141869{1} 0.256889{5} 0.287445{6} 0.220661{4}
    ˆα 0.3894{2} 0.390991{3} 0.41841{6} 0.365869{1} 0.416399{5} 0.406432{4} 0.44011{7}
    ˆλ 0.356816{1} 0.382947{3} 0.413734{6} 0.367721{2} 0.417675{7} 0.409102{5} 0.39411{4}
    MSE ˆθ 0.0465{7} 0.034473{3} 0.031461{1} 0.037084{6} 0.035224{4} 0.032308{2} 0.035791{5}
    ˆβ 0.065583{2} 0.070206{3} 0.158663{7} 0.036064{1} 0.125967{5} 0.146189{6} 0.087577{4}
    ˆα 0.209603{1} 0.225049{2} 0.244866{5} 0.233893{3} 0.263057{6} 0.24107{4} 0.278205{7}
    ˆλ 0.144604{1} 0.167324{3} 0.185792{6} 0.15515{2} 0.188622{7} 0.182576{5} 0.174821{4}
    MRE ˆθ 0.665755{7} 0.638559{3} 0.625369{1} 0.625954{2} 0.660147{6} 0.646518{4} 0.647238{5}
    ˆβ 0.40623{3} 0.402921{2} 0.604664{7} 0.283738{1} 0.513778{5} 0.574889{6} 0.441321{4}
    ˆα 0.5192{2} 0.521321{3} 0.557879{6} 0.487826{1} 0.555199{5} 0.541909{4} 0.586813{7}
    ˆλ 0.713633{1} 0.765894{3} 0.827467{6} 0.735443{2} 0.83535{7} 0.818203{5} 0.788219{4}
    Ranks 37{3} 33{2} 59{5} 24{1} 68{7} 55{4} 60{6}
    100 BIAS ˆσ 0.153797{7} 0.147122{5} 0.135032{3} 0.129218{1} 0.134948{2} 0.143487{4} 0.153338{6}
    ˆβ 0.120608{2} 0.128477{3} 0.186858{7} 0.086896{1} 0.164689{5} 0.171597{6} 0.136009{4}
    ˆα 0.377902{3} 0.386078{4} 0.403267{6} 0.288504{1} 0.392481{5} 0.367004{2} 0.407771{7}
    ˆλ 0.29636{1} 0.326781{3} 0.372342{7} 0.313049{2} 0.362218{6} 0.362159{5} 0.344262{4}
    MSE ˆθ 0.039643{7} 0.034054{5} 0.025299{2} 0.025729{3} 0.024872{1} 0.026762{4} 0.036139{6}
    ˆβ 0.024458{2} 0.026304{3} 0.056386{7} 0.012241{1} 0.045467{5} 0.048396{6} 0.031656{4}
    ˆα 0.214127{2} 0.234593{4} 0.250263{6} 0.175237{1} 0.240497{5} 0.220399{3} 0.254909{7}
    ˆλ 0.107776{1} 0.132118{3} 0.157282{7} 0.120781{2} 0.150874{5} 0.15114{6} 0.143373{4}
    MRE ˆθ 0.61519{7} 0.588486{5} 0.540129{3} 0.516873{1} 0.539792{2} 0.573948{4} 0.613352{6}
    ˆβ 0.241216{2} 0.256955{3} 0.373716{7} 0.173793{1} 0.329378{5} 0.343194{6} 0.272019{4}
    ˆα 0.503869{3} 0.514771{4} 0.537689{6} 0.384672{1} 0.523308{5} 0.489339{2} 0.543695{7}
    ˆλ 0.59272{1} 0.653562{3} 0.744684{7} 0.626099{2} 0.724437{6} 0.724317{5} 0.688523{4}
    Ranks 38{2} 45{3} 68{7} 17{1} 52{4} 53{5} 63{6}
    200 BIAS ˆσ 0.143248{6} 0.139091{5} 0.128434{3} 0.10969{1} 0.137393{4} 0.120289{2} 0.1452{7}
    ˆβ 0.081747{2} 0.085726{3} 0.113757{6} 0.060472{1} 0.108641{5} 0.116098{7} 0.089427{4}
    ˆα 0.343484{3} 0.352998{4} 0.353592{5} 0.233319{1} 0.369344{6} 0.333755{2} 0.38331{7}
    ˆλ 0.257837{1} 0.272929{3} 0.306274{6} 0.266796{2} 0.313787{7} 0.301849{5} 0.287142{4}
    MSE ˆθ 0.036058{7} 0.033323{5} 0.024352{3} 0.02014{1} 0.02754{4} 0.020168{2} 0.035779{6}
    ˆβ 0.011033{2} 0.011812{3} 0.021639{6} 0.00578{1} 0.019095{5} 0.021922{7} 0.01286{4}
    ˆα 0.182778{2} 0.209958{5} 0.20589{4} 0.139641{1} 0.23027{6} 0.185814{3} 0.231535{7}
    ˆλ 0.091114{1} 0.101621{3} 0.115374{5} 0.101162{2} 0.121723{7} 0.117581{6} 0.112911{4}
    MRE ˆθ 0.57299{6} 0.556365{5} 0.513737{3} 0.438759{1} 0.549572{4} 0.481154{2} 0.580799{7}
    ˆβ 0.163494{2} 0.171453{3} 0.227514{6} 0.120944{1} 0.217282{5} 0.232195{7} 0.178855{4}
    ˆα 0.457978{3} 0.470533{4} 0.471456{5} 0.311092{1} 0.492459{6} 0.445007{2} 0.51108{7}
    ˆλ 0.515674{1} 0.545859{3} 0.612548{6} 0.533592{2} 0.627574{7} 0.603699{5} 0.574284{4}
    Ranks 36{2} 46{3} 58{5} 15{1} 66{7} 50{4} 65{6}
    300 BIAS ˆσ 0.138561{5} 0.138807{6} 0.124072{3} 0.090879{1} 0.131573{4} 0.118318{2} 0.146928{7}
    ˆβ 0.058973{2} 0.067624{3} 0.090901{7} 0.050863{1} 0.084914{5} 0.088902{6} 0.069056{4}
    ˆα 0.337647{2} 0.350945{4} 0.360021{6} 0.193548{1} 0.359778{5} 0.346452{3} 0.369353{7}
    ˆλ 0.219299{1} 0.245087{3} 0.279345{6} 0.232013{2} 0.279926{7} 0.274285{5} 0.247637{4}
    MSE ˆθ 0.034373{6} 0.033063{5} 0.023835{3} 0.013914{1} 0.026133{4} 0.020005{2} 0.037861{7}
    ˆβ 0.005718{2} 0.007306{3} 0.013345{7} 0.004054{1} 0.011805{5} 0.012591{6} 0.007862{4}
    ˆα 0.191074{2} 0.214125{4} 0.222387{6} 0.109753{1} 0.225309{7} 0.208967{3} 0.220442{5}
    ˆλ 0.072871{1} 0.091013{3} 0.104698{6} 0.082986{2} 0.106309{7} 0.102139{5} 0.095991{4}
    MRE ˆθ 0.554242{5} 0.556827{6} 0.496287{3} 0.363517{1} 0.526293{4} 0.473271{2} 0.587712{7}
    ˆβ 0.117947{2} 0.135248{3} 0.181803{7} 0.101726{1} 0.169828{5} 0.177805{6} 0.138111{4}
    ˆα 0.450196{2} 0.471926{4} 0.480028{6} 0.258064{1} 0.479704{5} 0.461936{3} 0.492471{7}
    ˆλ 0.438598{1} 0.490174{3} 0.55869{6} 0.464026{2} 0.559852{7} 0.548569{5} 0.495274{4}
    Ranks 31{2} 47{3} 66{7} 15{1} 65{6} 48{4} 64{5}
    500 BIAS ˆσ 0.123165{3} 0.13065{6} 0.125115{4} 0.080862{1} 0.129143{5} 0.110391{2} 0.142231{7}
    ˆβ 0.04558{2} 0.050097{3} 0.068652{7} 0.041848{1} 0.065164{5} 0.065489{6} 0.051396{4}
    ˆα 0.304704{2} 0.3462{4} 0.346591{5} 0.156206{1} 0.36074{6} 0.345446{3} 0.372417{7}
    ˆλ 0.189126{1} 0.205681{3} 0.240779{6} 0.206657{4} 0.251445{7} 0.227929{5} 0.199507{2}
    MSE ˆθ 0.027603{5} 0.032787{6} 0.02505{3} 0.011429{1} 0.025908{4} 0.018361{2} 0.035584{7}
    ˆβ 0.003292{2} 0.004{3} 0.007501{7} 0.002583{1} 0.006854{5} 0.006865{6} 0.004201{4}
    ˆα 0.166447{2} 0.213746{5} 0.209538{4} 0.083822{1} 0.23977{6} 0.208212{3} 0.242504{7}
    ˆλ 0.059636{1} 0.070925{4} 0.086588{6} 0.07067{3} 0.094284{7} 0.07912{5} 0.066433{2}
    MRE ˆθ 0.492662{3} 0.542602{6} 0.500462{4} 0.323447{1} 0.516574{5} 0.441562{2} 0.568925{7}
    ˆβ 0.091159{2} 0.100195{3} 0.137304{7} 0.083696{1} 0.130329{5} 0.130978{6} 0.102792{4}
    ˆα 0.406272{2} 0.4616{4} 0.462121{5} 0.208274{1} 0.480986{6} 0.460594{3} 0.496556{7}
    ˆλ 0.378253{1} 0.411362{3} 0.481558{6} 0.413313{4} 0.50289{7} 0.455858{5} 0.399014{2}
    Ranks 26{2} 50{4} 64{6} 20{1} 68{7} 48{3} 60{5}

     | Show Table
    DownLoad: CSV
    Table 5.  Simulation values of BIAS, MSE and MRE for (σ=1.5, β=0.25, α=1.5, λ=0.75).
    n Est. Est. Par. MLE ADE CVME MPSE LSE RTADE WLSE
    50 BIAS ˆσ 2.368473{3} 2.686051{4} 4.986483{7} 0.924537{1} 4.967695{6} 2.141082{2} 3.185041{5}
    ˆβ 0.097271{2} 0.100905{3} 0.136041{7} 0.05979{1} 0.126029{5} 0.129388{6} 0.108112{4}
    ˆα 0.768542{2} 0.932529{6} 0.817627{4} 0.528023{1} 0.810726{3} 0.887373{5} 1.025206{7}
    ˆλ 0.519419{2} 0.597718{5} 0.560367{3} 0.496842{1} 0.578398{4} 0.622348{6} 0.635568{7}
    MSE ˆθ 20.346394{2} 23.821214{4} 322.421312{6} 4.01809{1} 501.265568{7} 22.158699{3} 40.016079{5}
    ˆβ 0.017265{2} 0.017372{3} 0.039424{7} 0.007731{1} 0.0332{6} 0.031762{5} 0.022171{4}
    ˆα 0.881246{2} 1.233622{6} 1.011954{4} 0.584107{1} 0.992128{3} 1.180926{5} 1.406141{7}
    ˆλ 0.341328{1} 0.411674{5} 0.382462{3} 0.345231{2} 0.39751{4} 0.439778{6} 0.45337{7}
    MRE ˆθ 1.578982{3} 1.790701{4} 3.324322{7} 0.616358{1} 3.311797{6} 1.427388{2} 2.123361{5}
    ˆβ 0.389083{2} 0.403621{3} 0.544163{7} 0.239161{1} 0.504116{5} 0.517553{6} 0.432447{4}
    ˆα 0.512361{2} 0.621686{6} 0.545085{4} 0.352015{1} 0.540484{3} 0.591582{5} 0.683471{7}
    ˆλ 0.692559{2} 0.796958{5} 0.747155{3} 0.662455{1} 0.771197{4} 0.829797{6} 0.847424{7}
    Ranks 24{2} 53{3} 60{6} 17{1} 56{4.5} 56{4.5} 70{7}
    100 BIAS ˆσ 1.503596{2} 2.181679{4} 2.409605{6} 0.503445{1} 2.327618{5} 1.52453{3} 2.41262{7}
    ˆβ 0.056812{2} 0.065851{3} 0.085008{6} 0.03857{1} 0.078473{5} 0.085975{7} 0.069386{4}
    ˆα 0.697288{2} 0.956184{6} 0.816886{3} 0.398195{1} 0.848384{4} 0.907708{5} 1.022525{7}
    ˆλ 0.483617{2} 0.561585{5} 0.526647{4} 0.421428{1} 0.52466{3} 0.581553{6} 0.583707{7}
    MSE ˆθ 7.260581{3} 11.729038{4} 18.781725{7} 1.165435{1} 18.232529{6} 5.523101{2} 15.078136{5}
    ˆβ 0.005806{2} 0.00713{3} 0.013695{7} 0.002865{1} 0.011771{5} 0.0128{6} 0.008196{4}
    ˆα 0.794167{2} 1.330844{6} 1.077834{3} 0.454167{1} 1.167044{4} 1.289307{5} 1.437883{7}
    ˆλ 0.320773{2} 0.393818{5} 0.348875{4} 0.274376{1} 0.347553{3} 0.406129{6} 0.4076{7}
    MRE ˆθ 1.002397{2} 1.454453{4} 1.606403{6} 0.33563{1} 1.551745{5} 1.016354{3} 1.608414{7}
    ˆβ 0.227249{2} 0.263404{3} 0.340033{6} 0.154279{1} 0.31389{5} 0.343899{7} 0.277546{4}
    ˆα 0.464859{2} 0.637456{6} 0.544591{3} 0.265463{1} 0.565589{4} 0.605139{5} 0.681683{7}
    ˆλ 0.644823{2} 0.74878{5} 0.702196{4} 0.561904{1} 0.699547{3} 0.775404{6} 0.778275{7}
    Ranks 29{2} 52{4} 57{5} 12{1} 50{3} 65{6} 71{7}
    200 BIAS ˆσ 1.037079{2} 1.792147{5} 1.793938{6} 0.22218{1} 1.722149{4} 1.18732{3} 2.132244{7}
    ˆβ 0.036793{2} 0.043914{3} 0.053356{6} 0.024479{1} 0.050413{5} 0.053529{7} 0.047812{4}
    ˆα 0.626596{2} 0.923064{5} 0.850873{3} 0.234961{1} 0.862918{4} 0.938707{6} 1.045254{7}
    ˆλ 0.424606{2} 0.509404{5} 0.495485{4} 0.337934{1} 0.470998{3} 0.51275{6} 0.518243{7}
    MSE ˆθ 3.318423{3} 6.496031{4} 9.106189{7} 0.224702{1} 7.988472{5} 2.720292{2} 9.025803{6}
    ˆβ 0.002296{2} 0.003097{3} 0.005123{7} 0.001004{1} 0.004715{5} 0.004757{6} 0.003752{4}
    ˆα 0.718876{2} 1.320814{5} 1.176936{3} 0.234086{1} 1.259898{4} 1.424838{6} 1.574473{7}
    ˆλ 0.271847{2} 0.352271{7} 0.336857{5} 0.205673{1} 0.300407{3} 0.336161{4} 0.349541{6}
    MRE ˆθ 0.691386{2} 1.194765{5} 1.195959{6} 0.14812{1} 1.148099{4} 0.791547{3} 1.421496{7}
    ˆβ 0.147173{2} 0.175657{3} 0.213425{6} 0.097916{1} 0.201652{5} 0.214117{7} 0.191247{4}
    ˆα 0.41773{2} 0.615376{5} 0.567249{3} 0.15664{1} 0.575279{4} 0.625805{6} 0.696836{7}
    ˆλ 0.566142{2} 0.679205{5} 0.660647{4} 0.450579{1} 0.627997{3} 0.683667{6} 0.69099{7}
    Ranks 25{2} 55{4} 60{5} 12{1} 49{3} 62{6} 73{7}
    300 BIAS ˆσ 0.934476{2} 1.652431{5} 1.698162{6} 0.163951{1} 1.648421{4} 1.131801{3} 1.790429{7}
    ˆβ 0.029234{2} 0.036177{3} 0.044418{7} 0.021807{1} 0.042904{5} 0.044382{6} 0.036583{4}
    ˆα 0.549292{2} 0.912171{5} 0.879001{3} 0.182861{1} 0.895503{4} 0.963101{6} 1.051865{7}
    ˆλ 0.376163{2} 0.480521{5} 0.450893{3} 0.320269{1} 0.464435{4} 0.485137{7} 0.483487{6}
    MSE ˆθ 2.944863{3} 5.125591{4} 6.998085{7} 0.104482{1} 6.214904{6} 2.282795{2} 5.749254{5}
    ˆβ 0.001448{2} 0.001952{3} 0.003519{7} 0.000764{1} 0.003149{5} 0.00318{6} 0.002026{4}
    ˆα 0.633235{2} 1.271483{3} 1.280931{4} 0.136351{1} 1.30316{5} 1.514906{6} 1.618684{7}
    ˆλ 0.235934{2} 0.32667{7} 0.288002{3} 0.202082{1} 0.302176{4} 0.315643{5} 0.320899{6}
    MRE ˆθ 0.622984{2} 1.101621{5} 1.132108{6} 0.1093{1} 1.098947{4} 0.754534{3} 1.193619{7}
    ˆβ 0.116935{2} 0.144707{3} 0.177672{7} 0.087228{1} 0.171617{5} 0.177528{6} 0.146332{4}
    ˆα 0.366195{2} 0.608114{5} 0.586{3} 0.121907{1} 0.597002{4} 0.642067{6} 0.701243{7}
    ˆλ 0.501551{2} 0.640695{5} 0.601191{3} 0.427026{1} 0.619247{4} 0.64685{7} 0.644649{6}
    Ranks 25{2} 53{3} 59{5} 12{1} 54{4} 63{6} 70{7}
    500 BIAS ˆσ 0.707979{2} 1.670229{6} 1.495733{5} 0.096022{1} 1.415746{4} 1.155813{3} 1.76218{7}
    ˆβ 0.022165{2} 0.027766{3} 0.032408{6} 0.016335{1} 0.03191{5} 0.034972{7} 0.029233{4}
    ˆα 0.468421{2} 0.955412{6} 0.916134{4} 0.105347{1} 0.891212{3} 0.948734{5} 1.03083{7}
    ˆλ 0.31123{2} 0.421891{4} 0.441501{6} 0.232734{1} 0.430158{5} 0.443041{7} 0.411027{3}
    MSE ˆθ 1.945207{2} 5.032233{6} 4.953797{5} 0.022692{1} 4.44341{4} 2.264562{3} 5.551723{7}
    ˆβ 0.000806{2} 0.00117{3} 0.00175{6} 0.000427{1} 0.001659{5} 0.001978{7} 0.00128{4}
    ˆα 0.520655{2} 1.411546{5} 1.382384{4} 0.031728{1} 1.346687{3} 1.49178{6} 1.534144{7}
    ˆλ 0.166973{2} 0.273881{4} 0.298537{7} 0.125512{1} 0.283187{5} 0.287671{6} 0.249034{3}
    MRE ˆθ 0.471986{2} 1.113486{6} 0.997155{5} 0.064014{1} 0.94383{4} 0.770542{3} 1.174786{7}
    ˆβ 0.088661{2} 0.111063{3} 0.129633{6} 0.065338{1} 0.127639{5} 0.139889{7} 0.116933{4}
    ˆα 0.312281{2} 0.636941{6} 0.610756{4} 0.070231{1} 0.594141{3} 0.632489{5} 0.68722{7}
    ˆλ 0.414974{2} 0.562522{4} 0.588668{6} 0.310312{1} 0.573544{5} 0.590721{7} 0.548036{3}
    Ranks 24{2} 56{4} 64{6} 12{1} 51{3} 66{7} 63{5}

     | Show Table
    DownLoad: CSV
    Table 6.  Simulation values of BIAS, MSE and MRE for (σ=0.75, β=1.5, α=0.75, λ=1.5).
    n Est. Est. Par. MLE ADE CVME MPSE LSE RTADE WLSE
    50 BIAS ˆσ 0.533007{3} 0.525515{2} 0.723204{5} 0.768352{7} 0.669541{4} 0.735192{6} 0.467968{1}
    ˆβ 0.696631{3} 0.678629{2} 0.919443{7} 0.314638{1} 0.820546{4} 0.845912{6} 0.83555{5}
    ˆα 0.373187{1} 0.38736{2} 0.468738{7} 0.430852{5} 0.459504{6} 0.424095{4} 0.400041{3}
    ˆλ 1.235272{3} 1.233919{2} 1.310685{7} 0.852181{1} 1.258193{4} 1.30097{6} 1.258798{5}
    MSE ˆθ 0.538055{2} 0.588999{3} 1.283818{6} 1.342864{7} 1.038856{5} 0.968285{4} 0.41345{1}
    ˆβ 0.925072{3} 0.894862{2} 1.938067{7} 0.240667{1} 1.571116{5} 1.852425{6} 1.243068{4}
    ˆα 0.181722{1} 0.208587{2} 0.279792{7} 0.258366{5} 0.271894{6} 0.223819{3} 0.225754{4}
    ˆλ 1.915234{4} 1.876275{3} 2.075645{6} 1.170304{1} 1.92684{5} 2.166459{7} 1.8586{2}
    MRE ˆθ 0.710676{3} 0.700686{2} 0.964271{5} 1.02447{7} 0.892721{4} 0.980255{6} 0.623957{1}
    ˆβ 0.464421{3} 0.45242{2} 0.612962{7} 0.209759{1} 0.547031{4} 0.563942{6} 0.557033{5}
    ˆα 0.497583{1} 0.516479{2} 0.624984{7} 0.574469{5} 0.612672{6} 0.56546{4} 0.533389{3}
    ˆλ 0.823515{3} 0.822612{2} 0.87379{7} 0.568121{1} 0.838795{4} 0.867313{6} 0.839198{5}
    Ranks 30{2} 26{1} 78{7} 42{4} 57{5} 64{6} 39{3}
    100 BIAS ˆσ 0.455966{3} 0.450245{2} 0.457172{4} 0.547988{6} 0.473082{5} 0.605352{7} 0.425103{1}
    ˆβ 0.419928{2} 0.425707{3} 0.644488{7} 0.222528{1} 0.576533{6} 0.483531{5} 0.481736{4}
    ˆα 0.367284{1} 0.384975{3} 0.430198{7} 0.380888{2} 0.409335{5} 0.405695{4} 0.412138{6}
    ˆλ 1.08248{2} 1.124127{3} 1.213522{7} 0.766751{1} 1.200679{6} 1.158919{5} 1.136341{4}
    MSE ˆθ 0.395709{2} 0.409277{3} 0.478773{4} 0.690754{7} 0.518087{5} 0.685493{6} 0.320726{1}
    ˆβ 0.335538{2} 0.340951{3} 0.832996{7} 0.105629{1} 0.682518{6} 0.582234{5} 0.420308{4}
    ˆα 0.194956{1} 0.222087{2} 0.26745{7} 0.24238{5} 0.240445{4} 0.229644{3} 0.267244{6}
    ˆλ 1.557265{2} 1.674066{4} 1.833067{5} 1.152476{1} 1.851618{6} 1.870229{7} 1.655101{3}
    MRE ˆθ 0.607955{3} 0.600327{2} 0.609563{4} 0.730651{6} 0.630776{5} 0.807137{7} 0.566804{1}
    ˆβ 0.279952{2} 0.283805{3} 0.429658{7} 0.148352{1} 0.384355{6} 0.322354{5} 0.321157{4}
    ˆα 0.489712{1} 0.5133{3} 0.573597{7} 0.507851{2} 0.545779{5} 0.540927{4} 0.549518{6}
    ˆλ 0.721653{2} 0.749418{3} 0.809015{7} 0.511167{1} 0.800453{6} 0.772612{5} 0.757561{4}
    Ranks 23{1} 34{2.5} 73{7} 34{2.5} 65{6} 63{5} 44{4}
    200 BIAS ˆσ 0.402774{4} 0.402374{3} 0.377238{1} 0.417957{6} 0.380139{2} 0.491154{7} 0.41559{5}
    ˆβ 0.245344{2} 0.275749{3} 0.406656{7} 0.149827{1} 0.368069{6} 0.321157{5} 0.304178{4}
    ˆα 0.360952{2} 0.374004{3} 0.40199{7} 0.322812{1} 0.385975{5} 0.380121{4} 0.390848{6}
    ˆλ 0.946432{2} 1.022321{3} 1.095529{7} 0.58019{1} 1.09242{6} 1.029604{4} 1.060048{5}
    MSE ˆθ 0.302033{4} 0.309022{5} 0.27983{1} 0.437871{6} 0.282055{2} 0.456174{7} 0.301549{3}
    ˆβ 0.111483{2} 0.138574{3} 0.319138{7} 0.042499{1} 0.251714{6} 0.23398{5} 0.169932{4}
    ˆα 0.208275{2} 0.228173{4} 0.250129{7} 0.195911{1} 0.238967{5} 0.214321{3} 0.247718{6}
    ˆλ 1.344375{2} 1.523857{3} 1.637673{7} 0.793748{1} 1.626174{6} 1.601072{4} 1.606258{5}
    MRE ˆθ 0.537032{4} 0.536498{3} 0.502984{1} 0.557276{6} 0.506853{2} 0.654872{7} 0.55412{5}
    ˆβ 0.163562{2} 0.183833{3} 0.271104{7} 0.099885{1} 0.24538{6} 0.214105{5} 0.202785{4}
    ˆα 0.48127{2} 0.498672{3} 0.535987{7} 0.430416{1} 0.514634{5} 0.506829{4} 0.521131{6}
    ˆλ 0.630955{2} 0.681547{3} 0.730352{7} 0.386793{1} 0.72828{6} 0.686402{4} 0.706698{5}
    Ranks 30{2} 39{3} 66{7} 27{1} 57{4} 59{6} 58{5}
    300 BIAS ˆσ 0.354316{3} 0.387339{6} 0.340595{2} 0.314435{1} 0.356816{4} 0.423753{7} 0.366491{5}
    ˆβ 0.213491{2} 0.220535{3} 0.304576{7} 0.114722{1} 0.298278{6} 0.251673{5} 0.238859{4}
    ˆα 0.347032{2} 0.371622{4} 0.387408{7} 0.300536{1} 0.376533{6} 0.351687{3} 0.37459{5}
    ˆλ 0.887239{2} 0.950702{3} 1.009539{6} 0.464702{1} 1.04187{7} 0.969555{4} 0.994709{5}
    MSE ˆθ 0.234554{4} 0.255777{5} 0.20785{1} 0.266827{6} 0.22302{2} 0.328572{7} 0.227776{3}
    ˆβ 0.080081{2} 0.080411{3} 0.163303{6} 0.02384{1} 0.16766{7} 0.125845{5} 0.096789{4}
    ˆα 0.202885{3} 0.225264{4} 0.239855{6} 0.195118{1} 0.23271{5} 0.196299{2} 0.250337{7}
    ˆλ 1.225221{2} 1.42731{3} 1.496599{6} 0.593533{1} 1.625164{7} 1.487977{5} 1.487614{4}
    MRE ˆθ 0.472421{3} 0.516452{6} 0.454126{2} 0.419247{1} 0.475754{4} 0.565004{7} 0.488654{5}
    ˆβ 0.142327{2} 0.147024{3} 0.20305{7} 0.076481{1} 0.198852{6} 0.167782{5} 0.15924{4}
    ˆα 0.462709{2} 0.495496{4} 0.516544{7} 0.400714{1} 0.502044{6} 0.468916{3} 0.499454{5}
    ˆλ 0.591493{2} 0.633801{3} 0.673026{6} 0.309801{1} 0.69458{7} 0.64637{4} 0.66314{5}
    Ranks 29{2} 47{3} 63{6} 17{1} 67{7} 57{5} 56{4}
    500 BIAS ˆσ 0.319615{3} 0.337612{5} 0.318942{2} 0.226821{1} 0.324151{4} 0.352445{7} 0.341859{6}
    ˆβ 0.152897{2} 0.161827{3} 0.232809{6} 0.084409{1} 0.239478{7} 0.199409{5} 0.183279{4}
    ˆα 0.330652{2} 0.34606{4} 0.358393{5} 0.226347{1} 0.373944{7} 0.338023{3} 0.364009{6}
    ˆλ 0.725414{2} 0.781757{3} 0.941042{7} 0.307944{1} 0.93105{6} 0.892556{5} 0.82945{4}
    MSE ˆθ 0.171668{2} 0.197562{6} 0.173391{3} 0.160199{1} 0.173924{4} 0.224749{7} 0.18564{5}
    ˆβ 0.037617{2} 0.043746{3} 0.093287{6} 0.014092{1} 0.098688{7} 0.072681{5} 0.0554{4}
    ˆα 0.193338{2} 0.210329{4} 0.213739{5} 0.138737{1} 0.233195{6} 0.20139{3} 0.237107{7}
    ˆλ 0.923751{2} 1.044112{3} 1.35526{6} 0.32208{1} 1.362393{7} 1.323174{5} 1.154496{4}
    MRE ˆθ 0.426153{3} 0.45015{5} 0.425255{2} 0.302429{1} 0.432201{4} 0.469927{7} 0.455812{6}
    ˆβ 0.101931{2} 0.107885{3} 0.155206{6} 0.056273{1} 0.159652{7} 0.132939{5} 0.122186{4}
    ˆα 0.44087{2} 0.461413{4} 0.477857{5} 0.301796{1} 0.498592{7} 0.450697{3} 0.485345{6}
    ˆλ 0.48361{2} 0.521171{3} 0.627361{7} 0.205296{1} 0.6207{6} 0.595038{5} 0.552967{4}
    Ranks 26{2} 46{3} 60{5} 12{1} 72{7} 60{5} 60{5}

     | Show Table
    DownLoad: CSV
    Table 7.  Simulation values of BIAS, MSE and MRE for (σ=1.5, β=2.5, α=1.5, λ=2.5).
    n Est. Est. Par. MLE ADE CVME MPSE LSE RTADE WLSE
    50 BIAS ˆσ 3.071302{6} 2.218067{3} 3.407058{7} 0.671389{1} 2.611467{5} 2.090763{2} 2.304991{4}
    ˆβ 1.161666{6} 0.855781{2} 1.143568{5} 0.476107{1} 0.984626{4} 1.279159{7} 0.918149{3}
    ˆα 1.44144{7} 1.217317{5} 0.840631{3} 0.35853{1} 0.762359{2} 1.190678{4} 1.245395{6}
    ˆλ 1.222605{5} 1.208087{4} 1.144089{2} 0.853295{1} 1.172626{3} 1.257115{6} 1.288674{7}
    MSE ˆθ 34.936513{4} 22.291755{2} 200.181787{7} 4.491111{1} 143.181451{6} 33.681688{3} 35.74143{5}
    ˆβ 2.481203{5} 1.364787{2} 2.711632{6} 0.495917{1} 2.17304{4} 3.112637{7} 1.572973{3}
    ˆα 2.860903{7} 2.563188{6} 1.400073{3} 0.335785{1} 1.18094{2} 2.271422{5} 2.196576{4}
    ˆλ 1.740274{5} 1.704963{4} 1.528254{2} 1.206369{1} 1.606718{3} 1.790363{6} 1.894384{7}
    MRE ˆθ 2.047535{6} 1.478711{3} 2.271372{7} 0.447592{1} 1.740978{5} 1.393842{2} 1.536661{4}
    ˆβ 0.464667{6} 0.342313{2} 0.457427{5} 0.190443{1} 0.39385{4} 0.511664{7} 0.36726{3}
    ˆα 0.576576{7} 0.486927{5} 0.336253{3} 0.143412{1} 0.304944{2} 0.476271{4} 0.498158{6}
    ˆλ 0.81507{5} 0.805391{4} 0.762726{2} 0.568863{1} 0.781751{3} 0.838077{6} 0.859116{7}
    Ranks 70{7} 43{3} 47{4} 18{1} 38{2} 60{5.5} 60{5.5}
    100 BIAS ˆσ 1.901263{7} 1.856695{5} 1.751922{4} 0.374069{1} 1.450007{3} 1.362212{2} 1.878976{6}
    ˆβ 0.611938{3} 0.603637{2} 0.727946{6} 0.327749{1} 0.648047{5} 0.82756{7} 0.637527{4}
    ˆα 1.248797{4} 1.294143{6} 0.739916{3} 0.236211{1} 0.701416{2} 1.258424{5} 1.382138{7}
    ˆλ 1.105131{4} 1.138658{5} 1.076957{2} 0.716659{1} 1.09175{3} 1.18777{7} 1.148389{6}
    MSE ˆθ 9.7488{4} 8.653993{3} 15.521894{7} 0.299087{1} 12.491631{6} 4.683966{2} 10.355477{5}
    ˆβ 0.684923{3} 0.626635{2} 1.037086{6} 0.181031{1} 0.794992{5} 1.163807{7} 0.726858{4}
    ˆα 2.40867{4} 2.765037{7} 1.246549{3} 0.16602{1} 1.139392{2} 2.610954{5} 2.76102{6}
    ˆλ 1.59288{4} 1.629489{5} 1.409891{2} 1.017869{1} 1.444826{3} 1.64664{6} 1.647357{7}
    MRE ˆθ 1.267509{7} 1.237797{5} 1.167948{4} 0.249379{1} 0.966671{3} 0.908142{2} 1.25265{6}
    ˆβ 0.244775{3} 0.241455{2} 0.291178{6} 0.1311{1} 0.259219{5} 0.331024{7} 0.255011{4}
    ˆα 0.499519{4} 0.517657{6} 0.295966{3} 0.094484{1} 0.280566{2} 0.50337{5} 0.552855{7}
    ˆλ 0.736754{4} 0.759105{5} 0.717971{2} 0.477772{1} 0.727833{3} 0.791847{7} 0.765593{6}
    Ranks 54{4} 56{5} 45{3} 12{1} 39{2} 65{6.5} 65{6.5}
    200 BIAS ˆσ 1.489438{5} 1.492649{6} 1.251639{4} 0.254702{1} 1.124335{3} 1.032508{2} 1.631366{7}
    ˆβ 0.390916{2} 0.417238{3} 0.477833{6} 0.239941{1} 0.454654{5} 0.529328{7} 0.44636{4}
    ˆα 1.22106{4} 1.243143{5} 0.814897{3} 0.165892{1} 0.762155{2} 1.261539{6} 1.40163{7}
    ˆλ 0.983139{2} 1.046513{5} 1.037743{4} 0.559441{1} 1.011137{3} 1.07927{7} 1.074943{6}
    MSE ˆθ 4.861412{5} 4.482581{4} 5.358438{6} 0.138289{1} 3.857502{3} 2.057535{2} 5.46302{7}
    ˆβ 0.254524{2} 0.277628{3} 0.416379{6} 0.098282{1} 0.369749{5} 0.460917{7} 0.319359{4}
    ˆα 2.354586{4} 2.456133{5} 1.470366{3} 0.090532{1} 1.335057{2} 2.586728{6} 2.783177{7}
    ˆλ 1.427093{4} 1.53981{6} 1.420709{3} 0.7676{1} 1.346116{2} 1.502002{5} 1.613239{7}
    MRE ˆθ 0.992959{5} 0.995099{6} 0.834426{4} 0.169801{1} 0.749556{3} 0.688339{2} 1.087577{7}
    ˆβ 0.156367{2} 0.166895{3} 0.191133{6} 0.095976{1} 0.181861{5} 0.211731{7} 0.178544{4}
    ˆα 0.488424{4} 0.497257{5} 0.325959{3} 0.066357{1} 0.304862{2} 0.504616{6} 0.560652{7}
    ˆλ 0.655426{2} 0.697675{5} 0.691829{4} 0.37296{1} 0.674091{3} 0.719513{7} 0.716629{6}
    Ranks 41{3} 56{5} 52{4} 12{1} 38{2} 64{6} 73{7}
    300 BIAS ˆσ 1.293435{5} 1.405394{6} 1.18707{4} 0.18816{1} 1.05779{3} 1.013853{2} 1.454799{7}
    ˆβ 0.301219{2} 0.317693{3} 0.398356{6} 0.193541{1} 0.376531{5} 0.449381{7} 0.345671{4}
    ˆα 1.191676{4} 1.271217{5} 0.90975{3} 0.115396{1} 0.826743{2} 1.298864{6} 1.383884{7}
    ˆλ 0.883532{2} 0.922988{3} 0.986854{5} 0.433424{1} 0.986945{6} 1.030789{7} 0.942097{4}
    MSE ˆθ 3.641669{5} 3.869187{7} 3.296032{4} 0.082717{1} 2.34463{3} 1.912821{2} 3.775563{6}
    ˆβ 0.143738{2} 0.164582{3} 0.263128{6} 0.066753{1} 0.216256{5} 0.318415{7} 0.184695{4}
    ˆα 2.285817{4} 2.498052{5} 1.718056{3} 0.058219{1} 1.473023{2} 2.614885{6} 2.718042{7}
    ˆλ 1.234697{2} 1.315659{3} 1.325717{4} 0.624418{1} 1.339643{5} 1.436265{7} 1.349436{6}
    MRE ˆθ 0.86229{5} 0.936929{6} 0.79138{4} 0.12544{1} 0.705193{3} 0.675902{2} 0.969866{7}
    ˆβ 0.120487{2} 0.127077{3} 0.159342{6} 0.077417{1} 0.150612{5} 0.179752{7} 0.138268{4}
    ˆα 0.476671{4} 0.508487{5} 0.3639{3} 0.046158{1} 0.330697{2} 0.519546{6} 0.553553{7}
    ˆλ 0.589021{2} 0.615326{3} 0.657902{5} 0.28895{1} 0.657963{6} 0.687193{7} 0.628065{4}
    Ranks 39{2} 52{4} 53{5} 12{1} 47{3} 66{6} 67{7}
    500 BIAS ˆσ 1.07503{3} 1.211186{6} 1.107084{5} 0.137255{1} 1.081579{4} 0.853268{2} 1.319397{7}
    ˆβ 0.228821{2} 0.256498{3} 0.317303{6} 0.137947{1} 0.323822{7} 0.314426{5} 0.266151{4}
    ˆα 1.163324{4} 1.3026{6} 1.047082{3} 0.06968{1} 1.010481{2} 1.188082{5} 1.359189{7}
    ˆλ 0.731742{2} 0.814465{4} 0.924263{6} 0.297312{1} 0.924078{5} 0.937208{7} 0.810166{3}
    MSE ˆθ 2.473549{5} 2.645847{6} 2.210423{4} 0.041337{1} 2.195286{3} 1.264111{2} 3.143376{7}
    ˆβ 0.082146{2} 0.09989{3} 0.142998{5} 0.036235{1} 0.151097{7} 0.146639{6} 0.110179{4}
    ˆα 2.25719{4} 2.625202{6} 2.12372{3} 0.014391{1} 2.075987{2} 2.308054{5} 2.657415{7}
    ˆλ 0.921386{2} 1.101723{3} 1.266424{5} 0.352731{1} 1.30509{6} 1.343484{7} 1.105024{4}
    MRE ˆθ 0.716687{3} 0.807457{6} 0.738056{5} 0.091503{1} 0.721053{4} 0.568845{2} 0.879598{7}
    ˆβ 0.091528{2} 0.102599{3} 0.126921{6} 0.055179{1} 0.129529{7} 0.12577{5} 0.106461{4}
    ˆα 0.46533{4} 0.52104{6} 0.418833{3} 0.027872{1} 0.404192{2} 0.475233{5} 0.543676{7}
    ˆλ 0.487828{2} 0.542977{4} 0.616175{6} 0.198208{1} 0.616052{5} 0.624805{7} 0.540111{3}
    Ranks 35{2} 56{4} 57{5} 12{1} 54{3} 58{6} 64{7}

     | Show Table
    DownLoad: CSV
    Table 8.  Simulation values of BIAS, MSE and MRE for (σ=0.5, β=1.5, α=2.5, λ=0.25).
    n Est. Est. Par. MLE ADE CVME MPSE LSE RTADE WLSE
    50 BIAS ˆσ 2.893599{5} 2.06015{3} 5.432695{7} 0.540312{1} 2.900305{6} 1.933509{2} 2.470322{4}
    ˆβ 0.487959{5} 0.407624{2} 0.622881{7} 0.257203{1} 0.471101{4} 0.566687{6} 0.463078{3}
    ˆα 1.520295{7} 1.387888{5} 1.238646{3} 0.505153{1} 1.157777 ^\left\{ { {2} } \right\} 1.348129 ^\left\{ { {4} } \right\} 1.402493 ^\left\{ { {6} } \right\}
    \hat{\lambda} 0.186321 ^\left\{ { {1} } \right\} 0.212294 ^\left\{ { {3} } \right\} 0.214501 ^\left\{ { {4} } \right\} 0.207453 ^\left\{ { {2} } \right\} 0.219211 ^\left\{ { {6} } \right\} 0.224804 ^\left\{ { {7} } \right\} 0.216843 ^\left\{ { {5} } \right\}
    MSE \hat{\theta} 31.399644 ^\left\{ { {4} } \right\} 18.588935 ^\left\{ { {2} } \right\} 456.281219 ^\left\{ { {7} } \right\} 3.394397 ^\left\{ { {1} } \right\} 199.087447 ^\left\{ { {6} } \right\} 28.001065 ^\left\{ { {3} } \right\} 39.865189 ^\left\{ { {5} } \right\}
    \hat{\beta} 0.46565 ^\left\{ { {3} } \right\} 0.334741 ^\left\{ { {2} } \right\} 0.862602 ^\left\{ { {7} } \right\} 0.149008 ^\left\{ { {1} } \right\} 0.507983 ^\left\{ { {5} } \right\} 0.685578 ^\left\{ { {6} } \right\} 0.480383 ^\left\{ { {4} } \right\}
    \hat{\alpha} 3.02738 ^\left\{ { {7} } \right\} 2.789617 ^\left\{ { {6} } \right\} 2.213728 ^\left\{ { {3} } \right\} 0.600863 ^\left\{ { {1} } \right\} 1.945237 ^\left\{ { {2} } \right\} 2.638491 ^\left\{ { {4} } \right\} 2.675085 ^\left\{ { {5} } \right\}
    \hat{\lambda} 0.045601 ^\left\{ { {1} } \right\} 0.052655 ^\left\{ { {3} } \right\} 0.052244 ^\left\{ { {2} } \right\} 0.053603 ^\left\{ { {4.5} } \right\} 0.055113 ^\left\{ { {6} } \right\} 0.057037 ^\left\{ { {7} } \right\} 0.053603 ^\left\{ { {4.5} } \right\}
    MRE \hat{\theta} 5.787197 ^\left\{ { {5} } \right\} 4.1203 ^\left\{ { {3} } \right\} 10.86539 ^\left\{ { {7} } \right\} 1.080624 ^\left\{ { {1} } \right\} 5.800611 ^\left\{ { {6} } \right\} 3.867018 ^\left\{ { {2} } \right\} 4.940644 ^\left\{ { {4} } \right\}
    \hat{\beta} 0.325306 ^\left\{ { {5} } \right\} 0.271749 ^\left\{ { {2} } \right\} 0.415254 ^\left\{ { {7} } \right\} 0.171469 ^\left\{ { {1} } \right\} 0.314067 ^\left\{ { {4} } \right\} 0.377791 ^\left\{ { {6} } \right\} 0.308719 ^\left\{ { {3} } \right\}
    \hat{\alpha} 0.608118 ^\left\{ { {7} } \right\} 0.555155 ^\left\{ { {5} } \right\} 0.495459 ^\left\{ { {3} } \right\} 0.202061 ^\left\{ { {1} } \right\} 0.463111 ^\left\{ { {2} } \right\} 0.539251 ^\left\{ { {4} } \right\} 0.560997 ^\left\{ { {6} } \right\}
    \hat{\lambda} 0.745285 ^\left\{ { {1} } \right\} 0.849174 ^\left\{ { {3} } \right\} 0.858004 ^\left\{ { {4} } \right\} 0.829814 ^\left\{ { {2} } \right\} 0.876843 ^\left\{ { {6} } \right\} 0.899215 ^\left\{ { {7} } \right\} 0.867372 ^\left\{ { {5} } \right\}
    \sum Ranks 53 ^\left\{ { {4} } \right\} 39 ^\left\{ { {2} } \right\} 56 ^\left\{ { {5} } \right\} 20.5 ^\left\{ { {1} } \right\} 52 ^\left\{ { {3} } \right\} 59 ^\left\{ { {7} } \right\} 56.5 ^\left\{ { {6} } \right\}
    100 BIAS \hat{\sigma} 1.71046 ^\left\{ { {4} } \right\} 1.543353 ^\left\{ { {3} } \right\} 2.563169 ^\left\{ { {7} } \right\} 0.324322 ^\left\{ { {1} } \right\} 1.761025 ^\left\{ { {5} } \right\} 1.348114 ^\left\{ { {2} } \right\} 1.842607 ^\left\{ { {6} } \right\}
    \hat{\beta} 0.271716 ^\left\{ { {3} } \right\} 0.254277 ^\left\{ { {2} } \right\} 0.365534 ^\left\{ { {7} } \right\} 0.14494 ^\left\{ { {1} } \right\} 0.303646 ^\left\{ { {5} } \right\} 0.344037 ^\left\{ { {6} } \right\} 0.276847 ^\left\{ { {4} } \right\}
    \hat{\alpha} 1.378955 ^\left\{ { {4} } \right\} 1.405776 ^\left\{ { {6} } \right\} 1.171646 ^\left\{ { {3} } \right\} 0.286991 ^\left\{ { {1} } \right\} 1.122888 ^\left\{ { {2} } \right\} 1.395265 ^\left\{ { {5} } \right\} 1.433145 ^\left\{ { {7} } \right\}
    \hat{\lambda} 0.165496 ^\left\{ { {1} } \right\} 0.191167 ^\left\{ { {3} } \right\} 0.199523 ^\left\{ { {5} } \right\} 0.179456 ^\left\{ { {2} } \right\} 0.204775 ^\left\{ { {7} } \right\} 0.204683 ^\left\{ { {6} } \right\} 0.19562 ^\left\{ { {4} } \right\}
    MSE \hat{\theta} 7.565284 ^\left\{ { {3} } \right\} 8.06365 ^\left\{ { {4} } \right\} 92.946325 ^\left\{ { {7} } \right\} 0.270218 ^\left\{ { {1} } \right\} 14.630448 ^\left\{ { {6} } \right\} 5.312179 ^\left\{ { {2} } \right\} 11.461603 ^\left\{ { {5} } \right\}
    \hat{\beta} 0.137481 ^\left\{ { {3} } \right\} 0.128691 ^\left\{ { {2} } \right\} 0.291286 ^\left\{ { {7} } \right\} 0.036228 ^\left\{ { {1} } \right\} 0.191226 ^\left\{ { {5} } \right\} 0.229521 ^\left\{ { {6} } \right\} 0.155722 ^\left\{ { {4} } \right\}
    \hat{\alpha} 2.642845 ^\left\{ { {4} } \right\} 2.981665 ^\left\{ { {6} } \right\} 2.146734 ^\left\{ { {3} } \right\} 0.297547 ^\left\{ { {1} } \right\} 1.933317 ^\left\{ { {2} } \right\} 2.973921 ^\left\{ { {5} } \right\} 3.020674 ^\left\{ { {7} } \right\}
    \hat{\lambda} 0.038979 ^\left\{ { {1} } \right\} 0.046183 ^\left\{ { {3} } \right\} 0.049215 ^\left\{ { {5} } \right\} 0.043477 ^\left\{ { {2} } \right\} 0.051193 ^\left\{ { {7} } \right\} 0.050262 ^\left\{ { {6} } \right\} 0.04685 ^\left\{ { {4} } \right\}
    MRE \hat{\theta} 3.420919 ^\left\{ { {4} } \right\} 3.086707 ^\left\{ { {3} } \right\} 5.126338 ^\left\{ { {7} } \right\} 0.648643 ^\left\{ { {1} } \right\} 3.522051 ^\left\{ { {5} } \right\} 2.696228 ^\left\{ { {2} } \right\} 3.685214 ^\left\{ { {6} } \right\}
    \hat{\beta} 0.181144 ^\left\{ { {3} } \right\} 0.169518 ^\left\{ { {2} } \right\} 0.243689 ^\left\{ { {7} } \right\} 0.096627 ^\left\{ { {1} } \right\} 0.20243 ^\left\{ { {5} } \right\} 0.229358 ^\left\{ { {6} } \right\} 0.184565 ^\left\{ { {4} } \right\}
    \hat{\alpha} 0.551582 ^\left\{ { {4} } \right\} 0.56231 ^\left\{ { {6} } \right\} 0.468658 ^\left\{ { {3} } \right\} 0.114797 ^\left\{ { {1} } \right\} 0.449155 ^\left\{ { {2} } \right\} 0.558106 ^\left\{ { {5} } \right\} 0.573258 ^\left\{ { {7} } \right\}
    \hat{\lambda} 0.661985 ^\left\{ { {1} } \right\} 0.764667 ^\left\{ { {3} } \right\} 0.798091 ^\left\{ { {5} } \right\} 0.717824 ^\left\{ { {2} } \right\} 0.819099 ^\left\{ { {7} } \right\} 0.818733 ^\left\{ { {6} } \right\} 0.78248 ^\left\{ { {4} } \right\}
    \sum Ranks 37 ^\left\{ { {2} } \right\} 45 ^\left\{ { {3} } \right\} 66 ^\left\{ { {7} } \right\} 15 ^\left\{ { {1} } \right\} 55 ^\left\{ { {4} } \right\} 59 ^\left\{ { {5.5} } \right\} 59 ^\left\{ { {5.5} } \right\}
    200 BIAS \hat{\sigma} 1.317507 ^\left\{ { {3} } \right\} 1.393451 ^\left\{ { {4} } \right\} 1.807273 ^\left\{ { {7} } \right\} 0.239625 ^\left\{ { {1} } \right\} 1.417874 ^\left\{ { {5} } \right\} 1.057839 ^\left\{ { {2} } \right\} 1.572981 ^\left\{ { {6} } \right\}
    \hat{\beta} 0.177342 ^\left\{ { {2} } \right\} 0.186692 ^\left\{ { {4} } \right\} 0.235386 ^\left\{ { {7} } \right\} 0.102189 ^\left\{ { {1} } \right\} 0.203973 ^\left\{ { {5} } \right\} 0.207919 ^\left\{ { {6} } \right\} 0.178855 ^\left\{ { {3} } \right\}
    \hat{\alpha} 1.343016 ^\left\{ { {4} } \right\} 1.506946 ^\left\{ { {7} } \right\} 1.217309 ^\left\{ { {3} } \right\} 0.180074 ^\left\{ { {1} } \right\} 1.15298 ^\left\{ { {2} } \right\} 1.397913 ^\left\{ { {5} } \right\} 1.506771 ^\left\{ { {6} } \right\}
    \hat{\lambda} 0.141567 ^\left\{ { {1} } \right\} 0.175726 ^\left\{ { {4} } \right\} 0.176276 ^\left\{ { {5} } \right\} 0.155976 ^\left\{ { {2} } \right\} 0.188878 ^\left\{ { {7} } \right\} 0.186961 ^\left\{ { {6} } \right\} 0.167679 ^\left\{ { {3} } \right\}
    MSE \hat{\theta} 4.217578 ^\left\{ { {3} } \right\} 5.275548 ^\left\{ { {4} } \right\} 11.273828 ^\left\{ { {7} } \right\} 0.096368 ^\left\{ { {1} } \right\} 7.968936 ^\left\{ { {6} } \right\} 2.855553 ^\left\{ { {2} } \right\} 7.32838 ^\left\{ { {5} } \right\}
    \hat{\beta} 0.055162 ^\left\{ { {2} } \right\} 0.062272 ^\left\{ { {3} } \right\} 0.113856 ^\left\{ { {7} } \right\} 0.017041 ^\left\{ { {1} } \right\} 0.086531 ^\left\{ { {6} } \right\} 0.082157 ^\left\{ { {5} } \right\} 0.062758 ^\left\{ { {4} } \right\}
    \hat{\alpha} 2.584102 ^\left\{ { {4} } \right\} 3.404447 ^\left\{ { {7} } \right\} 2.24374 ^\left\{ { {3} } \right\} 0.196756 ^\left\{ { {1} } \right\} 2.150459 ^\left\{ { {2} } \right\} 3.080793 ^\left\{ { {5} } \right\} 3.33868 ^\left\{ { {6} } \right\}
    \hat{\lambda} 0.030984 ^\left\{ { {1} } \right\} 0.041857 ^\left\{ { {5} } \right\} 0.041581 ^\left\{ { {4} } \right\} 0.036696 ^\left\{ { {2} } \right\} 0.046431 ^\left\{ { {7} } \right\} 0.04599 ^\left\{ { {6} } \right\} 0.039497 ^\left\{ { {3} } \right\}
    MRE \hat{\theta} 2.635013 ^\left\{ { {3} } \right\} 2.786902 ^\left\{ { {4} } \right\} 3.614547 ^\left\{ { {7} } \right\} 0.479251 ^\left\{ { {1} } \right\} 2.835749 ^\left\{ { {5} } \right\} 2.115678 ^\left\{ { {2} } \right\} 3.145963 ^\left\{ { {6} } \right\}
    \hat{\beta} 0.118228 ^\left\{ { {2} } \right\} 0.124461 ^\left\{ { {4} } \right\} 0.156924 ^\left\{ { {7} } \right\} 0.068126 ^\left\{ { {1} } \right\} 0.135982 ^\left\{ { {5} } \right\} 0.138613 ^\left\{ { {6} } \right\} 0.119237 ^\left\{ { {3} } \right\}
    \hat{\alpha} 0.537206 ^\left\{ { {4} } \right\} 0.602778 ^\left\{ { {7} } \right\} 0.486923 ^\left\{ { {3} } \right\} 0.07203 ^\left\{ { {1} } \right\} 0.461192 ^\left\{ { {2} } \right\} 0.559165 ^\left\{ { {5} } \right\} 0.602709 ^\left\{ { {6} } \right\}
    \hat{\lambda} 0.56627 ^\left\{ { {1} } \right\} 0.702902 ^\left\{ { {4} } \right\} 0.705102 ^\left\{ { {5} } \right\} 0.623904 ^\left\{ { {2} } \right\} 0.755514 ^\left\{ { {7} } \right\} 0.747844 ^\left\{ { {6} } \right\} 0.670717 ^\left\{ { {3} } \right\}
    \sum Ranks 31 ^\left\{ { {2} } \right\} 58 ^\left\{ { {5} } \right\} 60 ^\left\{ { {6.5} } \right\} 15 ^\left\{ { {1} } \right\} 60 ^\left\{ { {6.5} } \right\} 57 ^\left\{ { {4} } \right\} 55 ^\left\{ { {3} } \right\}
    300 BIAS \hat{\sigma} 1.144979 ^\left\{ { {3} } \right\} 1.146644 ^\left\{ { {4} } \right\} 1.531675 ^\left\{ { {7} } \right\} 0.193615 ^\left\{ { {1} } \right\} 1.305878 ^\left\{ { {5} } \right\} 0.901886 ^\left\{ { {2} } \right\} 1.35253 ^\left\{ { {6} } \right\}
    \hat{\beta} 0.136016 ^\left\{ { {3} } \right\} 0.131749 ^\left\{ { {2} } \right\} 0.185296 ^\left\{ { {7} } \right\} 0.084294 ^\left\{ { {1} } \right\} 0.164244 ^\left\{ { {6} } \right\} 0.153234 ^\left\{ { {5} } \right\} 0.139283 ^\left\{ { {4} } \right\}
    \hat{\alpha} 1.351959 ^\left\{ { {4} } \right\} 1.432239 ^\left\{ { {5} } \right\} 1.222492 ^\left\{ { {3} } \right\} 0.126176 ^\left\{ { {1} } \right\} 1.127881 ^\left\{ { {2} } \right\} 1.47839 ^\left\{ { {6} } \right\} 1.569137 ^\left\{ { {7} } \right\}
    \hat{\lambda} 0.135547 ^\left\{ { {2} } \right\} 0.173872 ^\left\{ { {6} } \right\} 0.169951 ^\left\{ { {4} } \right\} 0.129027 ^\left\{ { {1} } \right\} 0.177691 ^\left\{ { {7} } \right\} 0.170295 ^\left\{ { {5} } \right\} 0.15737 ^\left\{ { {3} } \right\}
    MSE \hat{\theta} 3.192105 ^\left\{ { {3} } \right\} 3.487222 ^\left\{ { {4} } \right\} 7.206876 ^\left\{ { {7} } \right\} 0.062511 ^\left\{ { {1} } \right\} 6.006768 ^\left\{ { {6} } \right\} 1.745369 ^\left\{ { {2} } \right\} 4.939705 ^\left\{ { {5} } \right\}
    \hat{\beta} 0.032751 ^\left\{ { {3} } \right\} 0.029968 ^\left\{ { {2} } \right\} 0.066741 ^\left\{ { {7} } \right\} 0.010959 ^\left\{ { {1} } \right\} 0.053713 ^\left\{ { {6} } \right\} 0.04456 ^\left\{ { {5} } \right\} 0.03511 ^\left\{ { {4} } \right\}
    \hat{\alpha} 2.684399 ^\left\{ { {4} } \right\} 3.26821 ^\left\{ { {5} } \right\} 2.300617 ^\left\{ { {3} } \right\} 0.099863 ^\left\{ { {1} } \right\} 2.089719 ^\left\{ { {2} } \right\} 3.416702 ^\left\{ { {6} } \right\} 3.431382 ^\left\{ { {7} } \right\}
    \hat{\lambda} 0.028587 ^\left\{ { {2} } \right\} 0.043915 ^\left\{ { {7} } \right\} 0.04007 ^\left\{ { {4} } \right\} 0.026484 ^\left\{ { {1} } \right\} 0.042333 ^\left\{ { {6} } \right\} 0.040364 ^\left\{ { {5} } \right\} 0.036334 ^\left\{ { {3} } \right\}
    MRE \hat{\theta} 2.289959 ^\left\{ { {3} } \right\} 2.293288 ^\left\{ { {4} } \right\} 3.06335 ^\left\{ { {7} } \right\} 0.38723 ^\left\{ { {1} } \right\} 2.611756 ^\left\{ { {5} } \right\} 1.803773 ^\left\{ { {2} } \right\} 2.705059 ^\left\{ { {6} } \right\}
    \hat{\beta} 0.090677 ^\left\{ { {3} } \right\} 0.087833 ^\left\{ { {2} } \right\} 0.123531 ^\left\{ { {7} } \right\} 0.056196 ^\left\{ { {1} } \right\} 0.109496 ^\left\{ { {6} } \right\} 0.102156 ^\left\{ { {5} } \right\} 0.092855 ^\left\{ { {4} } \right\}
    \hat{\alpha} 0.540784 ^\left\{ { {4} } \right\} 0.572895 ^\left\{ { {5} } \right\} 0.488997 ^\left\{ { {3} } \right\} 0.050471 ^\left\{ { {1} } \right\} 0.451152 ^\left\{ { {2} } \right\} 0.591356 ^\left\{ { {6} } \right\} 0.627655 ^\left\{ { {7} } \right\}
    \hat{\lambda} 0.542189 ^\left\{ { {2} } \right\} 0.695487 ^\left\{ { {6} } \right\} 0.679804 ^\left\{ { {4} } \right\} 0.516109 ^\left\{ { {1} } \right\} 0.710765 ^\left\{ { {7} } \right\} 0.681181 ^\left\{ { {5} } \right\} 0.629479 ^\left\{ { {3} } \right\}
    \sum Ranks 36 ^\left\{ { {2} } \right\} 52 ^\left\{ { {3} } \right\} 63 ^\left\{ { {7} } \right\} 12 ^\left\{ { {1} } \right\} 60 ^\left\{ { {6} } \right\} 54 ^\left\{ { {4} } \right\} 59 ^\left\{ { {5} } \right\}
    500 BIAS \hat{\sigma} 0.77681 ^\left\{ { {2} } \right\} 0.896938 ^\left\{ { {4} } \right\} 1.248395 ^\left\{ { {7} } \right\} 0.163887 ^\left\{ { {1} } \right\} 1.091826 ^\left\{ { {6} } \right\} 0.834705 ^\left\{ { {3} } \right\} 1.07752 ^\left\{ { {5} } \right\}
    \hat{\beta} 0.102285 ^\left\{ { {3} } \right\} 0.100908 ^\left\{ { {2} } \right\} 0.133641 ^\left\{ { {7} } \right\} 0.069737 ^\left\{ { {1} } \right\} 0.118822 ^\left\{ { {5} } \right\} 0.127745 ^\left\{ { {6} } \right\} 0.105786 ^\left\{ { {4} } \right\}
    \hat{\alpha} 1.315699 ^\left\{ { {4} } \right\} 1.505985 ^\left\{ { {6} } \right\} 1.177099 ^\left\{ { {2} } \right\} 0.074962 ^\left\{ { {1} } \right\} 1.218619 ^\left\{ { {3} } \right\} 1.48828 ^\left\{ { {5} } \right\} 1.531011 ^\left\{ { {7} } \right\}
    \hat{\lambda} 0.118643 ^\left\{ { {2} } \right\} 0.150958 ^\left\{ { {5} } \right\} 0.161525 ^\left\{ { {7} } \right\} 0.11325 ^\left\{ { {1} } \right\} 0.159578 ^\left\{ { {6} } \right\} 0.144845 ^\left\{ { {4} } \right\} 0.139587 ^\left\{ { {3} } \right\}
    MSE \hat{\theta} 1.414635 ^\left\{ { {2} } \right\} 1.855396 ^\left\{ { {4} } \right\} 4.459711 ^\left\{ { {7} } \right\} 0.045603 ^\left\{ { {1} } \right\} 3.281802 ^\left\{ { {6} } \right\} 1.495434 ^\left\{ { {3} } \right\} 3.14218 ^\left\{ { {5} } \right\}
    \hat{\beta} 0.018327 ^\left\{ { {3} } \right\} 0.017578 ^\left\{ { {2} } \right\} 0.03204 ^\left\{ { {7} } \right\} 0.007544 ^\left\{ { {1} } \right\} 0.024952 ^\left\{ { {5} } \right\} 0.028475 ^\left\{ { {6} } \right\} 0.020264 ^\left\{ { {4} } \right\}
    \hat{\alpha} 2.72946 ^\left\{ { {4} } \right\} 3.524665 ^\left\{ { {7} } \right\} 2.074643 ^\left\{ { {2} } \right\} 0.043589 ^\left\{ { {1} } \right\} 2.27618 ^\left\{ { {3} } \right\} 3.349662 ^\left\{ { {6} } \right\} 3.249412 ^\left\{ { {5} } \right\}
    \hat{\lambda} 0.022884 ^\left\{ { {2} } \right\} 0.033741 ^\left\{ { {5} } \right\} 0.038651 ^\left\{ { {7} } \right\} 0.022406 ^\left\{ { {1} } \right\} 0.037913 ^\left\{ { {6} } \right\} 0.03096 ^\left\{ { {4} } \right\} 0.029755 ^\left\{ { {3} } \right\}
    MRE \hat{\theta} 1.553621 ^\left\{ { {2} } \right\} 1.793875 ^\left\{ { {4} } \right\} 2.496789 ^\left\{ { {7} } \right\} 0.327773 ^\left\{ { {1} } \right\} 2.183652 ^\left\{ { {6} } \right\} 1.66941 ^\left\{ { {3} } \right\} 2.15504 ^\left\{ { {5} } \right\}
    \hat{\beta} 0.06819 ^\left\{ { {3} } \right\} 0.067272 ^\left\{ { {2} } \right\} 0.089094 ^\left\{ { {7} } \right\} 0.046492 ^\left\{ { {1} } \right\} 0.079215 ^\left\{ { {5} } \right\} 0.085163 ^\left\{ { {6} } \right\} 0.070524 ^\left\{ { {4} } \right\}
    \hat{\alpha} 0.52628 ^\left\{ { {4} } \right\} 0.602394 ^\left\{ { {6} } \right\} 0.47084 ^\left\{ { {2} } \right\} 0.029985 ^\left\{ { {1} } \right\} 0.487448 ^\left\{ { {3} } \right\} 0.595312 ^\left\{ { {5} } \right\} 0.612404 ^\left\{ { {7} } \right\}
    \hat{\lambda} 0.474573 ^\left\{ { {2} } \right\} 0.603834 ^\left\{ { {5} } \right\} 0.646098 ^\left\{ { {7} } \right\} 0.452999 ^\left\{ { {1} } \right\} 0.638314 ^\left\{ { {6} } \right\} 0.579381 ^\left\{ { {4} } \right\} 0.558348 ^\left\{ { {3} } \right\}
    \sum Ranks 33 ^\left\{ { {2} } \right\} 52 ^\left\{ { {3} } \right\} 69 ^\left\{ { {7} } \right\} 12 ^\left\{ { {1} } \right\} 60 ^\left\{ { {6} } \right\} 55 ^\left\{ { {4.5} } \right\} 55 ^\left\{ { {4.5} } \right\}

     | Show Table
    DownLoad: CSV

    In this section we will use all estimation methods presented in Section (4) with replacing out baseline model by Weibull distribution ( G\left(x, \xi \right) = 1-{\rm e}^{-\sigma\, x^{\beta}} ). Now, we will study the performance of the estimated parameters of the LocscW distribution by this estimation methods. Also, we do a comparison between all methods by using numerical values of average of bias (BIAS) |Bias(\widehat{\pmb \Delta})| = \frac{1}{M}\sum_{i = 1}^{M}|\widehat{\pmb \Delta}-\pmb \Delta| , mean squared errors (MSE), MSE = \frac{1}{M}\sum_{i = 1}^{M}(\widehat{\pmb \Delta}-\pmb \Delta)^2 , and mean relative errors (MRE) MRE = \frac{1}{M}\sum_{i = 1}^{M}|\widehat{\pmb \Delta}-\pmb \Delta|/\pmb \Delta , \pmb \Delta = (\alpha, \lambda, \xi) . The simulation results may be used to build and apply a guideline for choosing the best estimating approach for the specified model parameters. The R software (version 4.0.3) is used to produce M = 10,000 random samples from the proposed distribution for n = 50,100,200,300 and 500.

    The numerical results of simulations are reported in Tables 48 and the power of each value refers to its order in comparing all estimation methods with each other in the same line. Our estimators' partial and overall rankings are displayed in Table 9, in which we conclude that the best method for estimating proposed model parameters when having random samples from our model is MPSE, followed by MLE. Also, we found that as the sample increase, the absolute Bias and MSE and MRE diminishes

    Table 9.  Partial and overall ranks of all the methods of estimation of proposed distribution by various values of model parameters.
    Parameter n MLE ADE CVME MPSE OLSE RTADE WLSE
    \sigma=0.25, \beta=0.5, \alpha=0.75, \lambda=0.5 50 3.0 2.0 5.0 1.0 7.0 4.0 6.0
    100 2.0 3.0 7.0 1.0 4.0 5.0 6.0
    200 2.0 3.0 5.0 1.0 7.0 4.0 6.0
    300 2.0 3.0 7.0 1.0 6.0 4.0 5.0
    500 2.0 4.0 6.0 1.0 7.0 3.0 5.0
    \sigma=1.5, \beta=0.25, \alpha=1.5, \lambda=0.75 50 2.0 3.0 6.0 1.0 4.5 4.5 7.0
    100 2.0 4.0 5.0 1.0 3.0 6.0 7.0
    200 2.0 4.0 5.0 1.0 3.0 6.0 7.0
    300 2.0 3.0 5.0 1.0 4.0 6.0 7.0
    500 2.0 4.0 6.0 1.0 3.0 7.0 5.0
    \sigma=0.75, \beta=1.5, \alpha=0.75, \lambda=1.5 50 2.0 1.0 7.0 4.0 5.0 6.0 3.0
    100 1.0 2.5 7.0 2.5 6.0 5.0 4.0
    200 2.0 3.0 7.0 1.0 4.0 6.0 5.0
    300 2.0 3.0 6.0 1.0 7.0 5.0 4.0
    500 2.0 3.0 5.0 1.0 7.0 5.0 5.0
    \sigma=1.5, \beta=2.5, \alpha=1.5, \lambda=2.5 50 7.0 3.0 4.0 1.0 2.0 5.5 5.5
    100 4.0 5.0 3.0 1.0 2.0 6.5 6.5
    200 3.0 5.0 4.0 1.0 2.0 6.0 7.0
    300 2.0 4.0 5.0 1.0 3.0 6.0 7.0
    500 2.0 4.0 5.0 1.0 3.0 6.0 7.0
    \sigma=0.5, \beta=1.5, \alpha=2.5, \lambda=0.25 50 4.0 2.0 5.0 1.0 3.0 7.0 6.0
    100 2.0 3.0 7.0 1.0 4.0 5.5 5.5
    200 2.0 5.0 6.5 1.0 6.5 4.0 3.0
    300 2.0 3.0 7.0 1.0 6.0 4.0 5.0
    500 2.0 3.0 7.0 1.0 6.0 4.5 4.5
    \sum Ranks 60.0 82.5 142.5 29.5 115.0 131.5 139.0
    Overall Rank 2 3 7 1 4 5 6

     | Show Table
    DownLoad: CSV

    Researchers present two applications of the LocscW model in this section, one on hydrological data and the other on survival data. Using the approach of a limited-memory quasi-Newton code for bound-constrained optimization, we construct the log-likelihood function assessed at the MLEs ( \hat{\ell} ) (L-BFGS-B). We take into account many good statistics for model comparison, including the maximized log-likelihood ( \hat{\ell} ), Akaike information criterion (AIC), Corrected Akaike information criterion (CAIC), Bayesian information criterion (BIC), Hannan Quinn information criterion (HQIC), Anderson-Darling ( A^{*} ), Cramér–von Mises ( W^{*} ) and Kolmogorov-Smirnov (K-S) measures, where lower values of these statistics and higher p-values of K-S indicate good fits.

    The first data corresponds to the exceedances of flood peaks (in m3/s) of the Wheaton River near Carcross in Yukon Territory, Canada. They were analysed by Choulakian and Stephens (2001) and are listed below.

    1.7, 2.2, 14.4, 1.1, 0.4, 20.6, 5.3, 0.7, 1.9, 13.0, 12.0, 9.3, 1.4, 18.7, 8.5, 25.5, 11.6, 14.1, 22.1, 1.1, 2.5, 14.4, 1.7, 37.6, 0.6, 2.2, 39.0, 0.3, 15.0, 11.0, 7.3, 22.9, 1.7, 0.1, 1.1, 0.6, 9.0, 1.7, 7.0, 20.1, 0.4, 2.8, 14.1, 9.9, 10.4, 10.7, 30.0, 3.6, 5.6, 30.8, 13.3, 4.2, 25.5, 3.4, 11.9, 21.5, 27.6, 36.4, 2.7, 64.0, 1.5, 2.5, 27.4, 1.0, 27.1, 20.2, 16.8, 5.3, 9.7, 27.5, 2.5, 27.0.

    The summary statistics for these data are: n = 60, \bar{x} = 2.19297, s = 1.920062, skewness = 1.2614 and kurtosis = 2.23207.

    The histogram, box plot, and kernel density plots of the aforementioned data are shown in Figure 10, which demonstrates that the distribution is right-skewed, while the TTT plot is first convex and subsequently concave, indicating a bathtub failure rate. As a result, the LocscW distribution may theoretically be used to represent the existing data.

    Figure 10.  Histogram, TTT plot, box plot and Kernel density for Wheaton river data.

    Table 10 provides the MLEs of the LocscW parameters with standard errors (in parentheses) along with the competitor Weibull models. The outputs attest that 4 -parameter LocscW is the best fit because SEs are very small as compared to MLEs.

    Table 10.  MLEs and their standard errors (in parentheses) for Wheaton river data.
    Distribution a b c \alpha \beta \lambda
    LocscW 0.06168 1.3654 - 0.0600 - 0.1505
    (0.0051) (0.0066) - (0.0275) - (0.0234)
    EKumW 0.2267 0.5664 0.1461 10.6840 8.6820 -
    (0.0906) (0.0991) (0.0407) (0.2432) (0.1311) -
    McW 0.2109 1.2298 - 0.2703 0.1560 2.6487
    (0.0043) (0.0052) - (0.1034) (0.0229) (1.2659)
    BurrW 0.2073 0.2261 - 60.3845 4.0139 -
    (5.9539) (4.1151) - (140.5648) (73.0488) -
    BW 0490 1.2944 - 0.5511 0.4996 -
    (0.1305) (0.4677) - (0.2827) (0.6805) -
    GoW 0.1638 0.7765 0.2871 - 0.7264 -
    (2.5000) (0.1420) (4.4033) - (11.0908) -
    EOW 0.2597 0.5971 - 2.1902 0.6213 -
    (0.3080) (0.2603) - (2.0339) (0.3781) -
    GW 0.0144 1.3645 - - 0.5392 -
    (0.0227) (0.3784) - - (0.2152) -
    MOW 0.1261 0.8759 - - 1.1534 -
    (0.1074) (0.1686) - - (0.9517) -
    OLLW 0.0353 1.3800 - - 0.5914 -
    (0.0270) (0.2800) - - (1.4811) -
    LoW 0.3306 0.5792 - - 2.0934 -
    (6.6968) (10.6015) - - (38.3131) -
    W 0.1096 0.9011 - - - -
    (0.0301) (0.0855) - - - -

     | Show Table
    DownLoad: CSV

    Table 11 provides the values of AIC, CAIC, BIC, HQIC, A^{\ast} , W^{\ast} , K-S, and P-values for each model. We utilized all criteria of gof, and on the basis of these statistics outputs, the best fit model is LocscW (4-parameters only) than competitor Weibull models (having a greater number of parameters) and has the potential to fit right-skewed data with the bathtub failure rate.

    Table 11.  The statistics \hat\ell , AIC, CAIC, BIC, HQIC, A^{*} , W^{*} , K-S statistic and P-value for Wheaton river data.
    Distribution \hat\ell AIC CAIC BIC HQIC A^{*} W^{*} K-S P -value
    LocscW 247.9148 503.8296 504.4266 510.9363 507.4550 0.2832 0.0458 0.0579 0.9694
    EKumW 250.2966 510.5932 511.5023 521.9766 515.1250 0.5712 0.0961 0.0998 0.4696
    McW 249.4471 508.8943 509.8034 520.2776 513.4260 0.4876 0.0802 0.0947 0.5383
    BurrW 251.5910 511.1820 511.7790 520.2886 514.8073 0.8004 0.1408 0.1053 0.4018
    BW 250.9795 509.9589 510.5560 519.0656 513.5843 0.6339 0.1031 0.1074 0.3770
    GoW 250.7828 509.5656 510.1626 518.6722 513.1909 0.6201 0.1029 0.1037 0.4206
    EOW 250.7625 509.5250 510.1220 518.6317 513.1504 0.6023 0.0981 0.1053 0.4014
    GW 251.0818 508.1636 508.5165 514.9936 510.8826 0.6654 0.1103 0.1075 0.3764
    MOW 251.4838 508.9675 509.3205 515.7975 511.6866 0.7762 0.1355 0.1071 0.3806
    OLLW 249.9261 505.8522 506.2052 512.6822 508.5713 0.4738 0.0745 0.0962 0.5177
    LoW 257.8391 521.6782 522.0311 528.5082 524.3972 1.6484 0.2961 0.1138 0.3090
    W 251.4986 506.9973 507.1712 511.5506 508.8100 0.7855 0.1379 0.1052 0.4032

     | Show Table
    DownLoad: CSV

    The plots of the estimated densities are shown in Figure 11 while the plots of the estimated densities are shown in Figure 12 for estimated distribution functions for Wheaton river data.

    Figure 11.  The density plots of LocscW and competitive models using Wheaton river data.
    Figure 12.  The cdf plots of LocscW and competitive models using Wheaton river data.

    Figure 13 presents the plots of the estimated density in 13(a) while Figure 13(b) shows the plots the estimated cdf for LocscW model using Wheaton river data.

    Figure 13.  (a) Estimated density (b) Estimated cdf for LocscW model and other competitive models for Wheaton river data.

    Figure 14 presents the plot of the estimated density in 14(a) while Figure 14(b) shows the plot the estimated cdf and Figure 14(c) shows the P-P plot for LocscW model using Wheaton river data.

    Figure 14.  (a) Estimated density (b) Estimated cdf (c) P-P plot for the LocscW model for Wheaton river data.

    The next set of data reflects the survival times (in years) of a number of patients administered chemotherapy by Bekker et al. (2000). This data set's 47 values are as follows:

    0.047, 0.115, 0.121, 0.132, 0.164, 0.197, 0.203, 0.260, 0.282, 0.296, 0.334, 0.395, 0.458, 0.466, 0.501, 0.507, 0.529, 0.534, 0.540, 0.641, 0.644, 0.696, 0.841, 0.863, 1.099, 1.219, 1.271, 1.326, 1.447, 1.485, 1.553, 1.581, 1.589, 2.178, 2.343, 2.416, 2.444, 2.825, 2.830, 3.578, 3.658, 3.743, 3.978, 4.003, 4.033.

    The histogram, box plot, and kernel density of the above data are displayed in Figure 15 indicates that the distribution is right-skewed and unimodal while the TTT plot of the data is first convex and then concave (increasing-decreasing-increasing (confused type)), which suggests a model with heavy right tail is required, motivating the use of the LocscW model on these data.

    Figure 15.  Histogram, TTT plot, box plot and Kernel density for survival data.

    Table 12 provides the MLEs of the LocscW parameters with standard errors (in parentheses) along with the competitor Weibull models. The outputs attest that 4 -parameter LocscW is the best fit because SEs are very small as compared to MLEs.

    Table 12.  MLEs and their standard errors (in parentheses) for survival data.
    Distribution a b c \alpha \beta \lambda
    LocscW 0.2007 2.4503 - 0.1914 - 0.0064
    (0.1344) (0.4919) - (0.0673) - (0.0099)
    EKumW 0.0044 3.1433 8.9653 0.0596 2.0517 -
    (0.0050) (1.0831) (11.9282) (0.0.0681) (1.0049) -
    McW 2.7531 1.7413 - 0.0755 0.8566 10.9888
    (0.0440) (0.0383) - (0.0138) (0.0167) (0.1093)
    BurrW 0.2652 0.3311 - 51.2885 3.2075 -
    (3.6580) (3.4383) - (188.1357) (33.2955) -
    BW 9.3525 0.8476 - 2.6799 0.0987 -
    (0.0445) (0.0313) - (1.1839) (0.0158) -
    GoW 0.2984 0.9762 0.4072 - 2.1798 -
    (13.3466) (0.1949) (18.2412) - (97.5099) -
    EOW 1.3251 0.3423 - 5.3644 0.7698 -
    (1.5282) (0.2558) - (10.2286) (0.8812) -
    GW 1.9216 0.6738 - - 2.1270 -
    (3.6782) (0.6229) - - (3.4430) -
    MOW 0.4151 1.2322 - - 0.4432 -
    (0.3069) (0.2473) - - (0.4452) -
    OLLW 0.6269 1.4469 - - 0.6894 -
    (0.2227) (0.5643) - - (0.3085) -
    LoW 1.3361 1.5872 - - 0.9498 -
    (62.1457) (254.8173) - - (152.4854) -
    W 0.7178 1.0531 - - - -
    (0.1249) (0.1238) - - - -

     | Show Table
    DownLoad: CSV

    Table 13 provides the values of AIC, CAIC, BIC, HQIC, A^{\ast} , W^{\ast} , K-S, and P-values for each model. We utilized all criteria of gof, and on the basis of these statistics outputs, the best fit model is LocscW (4-parameters only) than competitor Weibull models (having a greater number of parameters) and has the potential to fit right-skewed data.

    Table 13.  The statistics \hat\ell , AIC, CAIC, BIC, HQIC, A^{*} , W^{*} , K-S statistic and P-value for survival data.
    Distribution \hat\ell AIC CAIC BIC HQIC A^{*} W^{*} K-S P -value
    LocscW 54.9274 117.8549 118.8549 125.0815 120.5489 0.2274 0.0284 0.0683 0.9752
    EKumW 57.0848 124.1697 125.7081 133.2030 127.5372 0.4797 0.0709 0.1035 0.6815
    McW 55.4733 120.9466 122.4851 129.9799 124.3142 0.2562 0.0301 0.0761 0.9390
    BurrW 58.1607 124.3214 125.3214 131.5481 127.0154 0.5389 0.0805 0.1089 0.6207
    BW 57.4021 122.8042 123.8042 130.0308 125.4982 0.3579 0.0491 0.0799 0.9141
    GoW 57.9819 123.9639 124.9639 131.1905 126.6579 0.5951 0.0896 0.1139 0.5838
    EOW 57.8992 123.7984 124.7984 131.0251 126.4924 0.4703 0.0691 0.1015 0.7049
    GW 57.9877 121.9756 122.5609 127.3955 123.9961 0.4525 0.0660 0.0977 0.7465
    MOW 57.8094 121.6188 122.2042 127.0388 123.6394 0.4599 0.0674 0.0961 0.7639
    OLLW 57.8423 121.6847 122.2700 127.1047 123.7052 0.6352 0.0957 0.1234 0.4631
    LoW 60.1856 126.3713 126.9566 131.7913 128.3918 0.5595 0.0808 0.0849 0.8746
    W 58.1237 120.2474 120.5331 126.8608 121.5944 0.5436 0.0813 0.1094 0.6149

     | Show Table
    DownLoad: CSV

    The plots of the estimated densities are shown in Figure 16, while the plots of the estimated CDFs for the LocscW model and its competing models utilizing survival data are shown in Figure 17.

    Figure 16.  The density plots of LocscW and competitive models using survival data.
    Figure 17.  The distribution function plots of LocscW and competitive models using survival data.

    Figure 18 presents the plots of the estimated density in 18(a) while Figure 18(b) shows the plots the estimated cdf for LocscW model for survival data.

    Figure 18.  (a) Estimated density(b)Estimated cdf for the LocscW model and other competitive models for survival data.

    Figure 19 presents the plot of the estimated density in 19(a) while Figure 19(b) shows the plot the estimated cdf and Figure 19(c) shows the P-P plot for LocscW model using Wheaton river data.

    Figure 19.  (a) Estimated density (b) Estimated cdf (c) P-P plot for the LocscW model for survival data.

    In this paper, we presented a new lomax-G family of distributions using odd sine/cosecant function (Locsc-G) and obtained prominent mathematical properties such as reliability functions, linear representation for cdf and pdf in terms of exp-G distributions, ordinary and weighted moments, quantile and moment generating function, stress-strength reliability, stochastic ordering, and order statistics. Using well-known distributions, the graphical analysis is performed to observe the flexibility in the proposed family with almost all unimodal shapes of densities and hazard rate functions. Moreover, a new Lomax cosecant Weibull distribution (LocscW), a four-parameter model, is also discussed in detail. The model parameters are estimated by the method of maximum likelihood. We used almost all goodness-of-fit criteria to prove the usefulness of the proposed family and model (LocscW) by means of applications of two data sets. We forecast the wider utility of the new family and model in statistical fields, chiefly in hydrological studies, survival analysis, and reliability engineering.

    This research was supported by Researchers Supporting Project number (RSP-2021/156), King Saud University, Riyadh, Saudi Arabia.

    The authors declare there is no conflict of interest.



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