Research article Special Issues

Predictive modeling of reliability engineering data using a new version of the flexible Weibull model


  • Received: 11 December 2022 Revised: 04 February 2023 Accepted: 20 February 2023 Published: 27 March 2023
  • The combined-unified hybrid sampling approach was introduced as a general model that combines the unified hybrid censoring sampling approach and the combined hybrid censoring approach into a unified approach. In this paper, we apply this censoring sampling approach to improve the estimation of the parameter via a novel five-parameter expansion distribution, which we call the generalized Weibull-modified Weibull model. The new distribution contains five parameters and is therefore very flexible in terms of accommodating different types of data. The new distribution provides graphs of the probability density function, e.g., symmetric or right skewed. The graph of the risk function can have a shape similar to a monomer of the increasing or decreasing model. Using the Monte Carlo method, the maximum likelihood approach is used in the estimation procedure. The Copula model was used to discuss the two marginal univariate distributions. The asymptotic confidence intervals of the parameters were developed. We present some simulation results to validate the theoretical results. Finally, a data set with failure times for 50 electronic components was analyzed to illustrate the applicability and potential of the proposed model.

    Citation: Walid Emam, Ghadah Alomani. Predictive modeling of reliability engineering data using a new version of the flexible Weibull model[J]. Mathematical Biosciences and Engineering, 2023, 20(6): 9948-9964. doi: 10.3934/mbe.2023436

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  • The combined-unified hybrid sampling approach was introduced as a general model that combines the unified hybrid censoring sampling approach and the combined hybrid censoring approach into a unified approach. In this paper, we apply this censoring sampling approach to improve the estimation of the parameter via a novel five-parameter expansion distribution, which we call the generalized Weibull-modified Weibull model. The new distribution contains five parameters and is therefore very flexible in terms of accommodating different types of data. The new distribution provides graphs of the probability density function, e.g., symmetric or right skewed. The graph of the risk function can have a shape similar to a monomer of the increasing or decreasing model. Using the Monte Carlo method, the maximum likelihood approach is used in the estimation procedure. The Copula model was used to discuss the two marginal univariate distributions. The asymptotic confidence intervals of the parameters were developed. We present some simulation results to validate the theoretical results. Finally, a data set with failure times for 50 electronic components was analyzed to illustrate the applicability and potential of the proposed model.



    There are many phenomena in this world that need statistical description to be more understandable to the reader, but there is no specific statistical distribution that describes all of them. Therefore, many researchers have recently tried to develop new families by adding one, two or three parameters, e.g., [1,2,3,4,5]. By adding two additional parameters β and γ, Cordeiro et al. [6] introduced the generalized Weibull distribution family. We continue this line of research by proposing a novel family, namely, a generalized Weibull-modified Weibull model. A model was developed that can fit real data because the model has great flexibility in representing nonlinear dynamics. The model is useful for studying data science through statistical modeling and it has been applied to engineering data. Its application in engineering shows that the developed model is adaptable and flexible in terms of ability to represent complex data. For any cumulative function (CDF) W(x) and probability density function (PDF) w(x), the CDF and the PDF of the proposed family are respectively given by

    ϝ(x;β,γ)=1exp[β(log[1W(x)])γ],xR;β,γ>0, (1.1)

    and

    f(x;β,γ)=βγw(x)1W(x)(log[1W(x)])γ1exp(β(log(1W(x)))γ),xR;β,γ>0. (1.2)

    Type-Ⅰ and Type-Ⅱ-censorship schemes are the two most common and popular censorship schemes. Type-Ⅰ and type-Ⅱ censorship schemes were merged by Epstein [7] in the hybrid censorship scheme. For a more brief review of censoring schemes, we refer the reader to [8–14]. Balakrishnan et al. [15] have proposed a unified hybrid censoring method. Huang and Yang [16] have considered a combined hybrid censoring sample. Emam and Sultan [17] combined the unified hybrid censoring sampling method and the combined hybrid censoring sampling method into a unified approach known as C-UHCS(m,r;T1,T2), which refers to the combined-unified hybrid censoring method. The likelihood function of C-UHCS(k,r;T1,T2) is

    L(Ω|xk)=n!(nk)![1F(T)]nkki=1f(xi), (1.3)

    where k and T can be chosen as:

    L(C)(Ω|x) L(U)(Ω|x)
    Cases k T k T
    1 :  0<T1<Xk:n<T2<Xr:n m Xm:n D2 T2
    2 :  0<T1<Xk:n<Xr:n<T2 m Xm:n r Xr:n
    3 :  0<T1<T2<Xk:n<Xr:n D2 T2 m Xm:n
    4 :  0<Xk:n<Xr:n<T1<T2 r Xr:n D1 T1
    5 :  0<Xk:n<T1<Xr:n<T2 D1 T1 r Xr:n
    6 :  0<Xk:n<T1<T2<Xr:n D1 T1 D2 T2

    Then, for a parameter space Ω, the likelihood function of C-UHCS(m,r;T1,T2), which represents all possible likelihood functions under different values of k, T, and xk=(x1,x2,...,xk), can be written as

    L(Ω|xk)=n!(nk)!(ki=1f(xi))(1F(T))nk. (1.4)

    The authors believe that this problem deserves investigation. The main motivations for using the GMW-X family in practice are as follows: 1) It is an excellent way to enter additional parameters to create an extended version of the basic model. 2) It can improve the properties of the traditional distributions. 3) It can create symmetric, right-skewed and left-skewed distributions. 4) It can provide a consistently better fit than other models. This was a good incentive to study the problem, and this was supported by the numerical results, which confirmed the superiority of the new model over many of the basic and competing models.

    The rest of this work is presented here as follows. The generalized Weibull-modified Weibull distribution (GWMWD) is presented in Section 2. The bivariate extension of the generalized Weibull-modified Weibull model is discussed in Section 3. Based on C-UHCS(m,r;T1,T2), Section 4 is devoted to applying the maximum likelihood approach to the GWMWD. Section 5 presents the Monte Carlo procedure. Section 6 applies the GWMWD to a data set of 50 electronic component failures. Section 7 shows some conclusions.

    Let X be a random variable (R.V.) with the modified Weibull distribution (ω,θ,ν) distribution suggested by Sarhan and Zaindin [18]; then, its CDF is

    W(x;ω,δ,ν)=1exp(ωxδxν),x>0, (2.1)

    and its PDF is given by

    w(x;ω,δ,ν)=(ω+δνxν1)exp(ωxδxν),x>0, (2.2)

    where ω0 is a scale parameter, while δ0 and ν>0 are shape parameters such that ν+δ>0.

    The generalized Weibull distribution family generalizes the generalized Weibull normal distribution when β=γ=1, the generalized Weibull Gumbel distribution when the generalized Weibull family βγ=1, and the generalized Weibull logistic distribution. The GWMWD is defined from (1.1) by replacing W(x) and w(x) with W(x;ω,δ,ν) and w(x;ω,δ,ν), respectively. The CDF and PDF of GWMWD, respectively, are

    ϝ(x;β,γ,ω,δ,ν)=1eβ(xω+xνδ)γ,x>0;β,γ,ω,δ,ν>0, (2.3)

    and

    f(x;β,γ,ω,δ,ν)=eβ(xω+xνδ)γβγ(ω+xν1δν)(xω+xνδ)γ1,x>0;β,γ,ω,δ,ν>0. (2.4)

    The survival function (SF) and hazard rate function (HRF) via the GWMWD of time t, respectively, are

    S(t;β,γ,ω,δ,ν)=eβ(tω+tνδ)γ, (2.5)

    and

    H(t;β,γ,ω,δ,ν)=βγ(tω+tνδ)γ1(ω+tν1δν). (2.6)

    In particular, the GWMWD generalizes the generalized Weibull-Weibull distribution (when ω=0), the generalized Weibull-Rayleigh distribution (when ω=0 and ν=2), the generalized linear Weibull exponential distribution (when ν=2 and δ=ω/2, ω>0), and the generalized Weibull exponential distribution (for ν=0). In what follows, an R.V. X with the GWMWD PDF (2.3) is written as XGWMWD(β,γ,ω,δ,ν).

    Some possible behaviors of the CDF, PDF, and HRF for GWMWD (β,γ,ω,δ,ν) are shown in Figure 1. The left panel shows GWMWD (10.2, 10.9, 0.50, 12.9, 4.20), GWMWD(0.20, 0.90, 0.50, 1.90, 0.20), GWMWD (10.20, 6.90, 0.50, 1.90, 0.20), GWMWD (1.20, 3.90, 0.65, 0.90, 0.20), and GWMWD (0.20, 12.9, 0.30, 1.10, 0.01), while the right panel shows GWMWD (10.2, 10.9, 0.50, 12.9, 4.20), GWMWD (7.20, 8.90, 1.50, 12.9, 4.20), GWMWD (8.20, 7.90, 3.90, 10.9, 3.50), GWMWD (8.20, 6.90, 7.50, 12.9, 4.20), and GWMWD (1.20, 9.90, 2.50, 12.9, 4.20). From Figure 1, it can be seen that the CDF increases faster with increasing x for the parameters β and δ>1; then, it is constant and the graph grows exponentially, the PDF increases faster with increasing x for the parameters β and δ>1, and the proposed GWMWD has strong spurs; the HRF is constant and then increases faster with increasing x when the parameters β,γ,δ and α>1. The PDF shape is at times very flexible. It appears to approximate a bell curve with some twist. At other times it appears to have strong tails. Because of the divergent behavior of the proposed model, it could be a good candidate for modeling semi-normal and strong-tailed data in various industrial, financial, and medical applications.

    Figure 1.  Plots for different CDFs, PDFs and HRFs for GWMWD (β,γ,ω,δ,ν).

    The copula model was introduced by Morgenstern [19] to represent the joint CDF of the two marginal univariate distributions. Let ϝ(xj) be the CDF of Xj,j=1,2. Conway [20] introduced the joint CDF and PDF of the copula model, respectively, as

    ϝ(x1,x2)=ϝ(x1)ϝ(x2)[1+ρ(1ϝ(x1))(1ϝ(x2))],1<ρ<1, (3.1)

    and

    f(x1,x2)=f(x1)f(x2)[1+ρ(12ϝ(x1))(12ϝ(x2))], (3.2)

    where ρ is the dependence measure between X1 and X2. Let the R.V.s X1 GWMWD (β1,γ1,ω1,δ1,ν1) and X2 GWMWD (β2,γ2,ω2,δ2,ν2); then, the corresponding joint CDF and PDF are, respectively, given by

    ϝ(x1,x2)=(1eβ1(x1ω1+xν11δ1)γ1)(1eβ2(x2ω2+xν22δ2)γ2)×(1+ρeβ1(x1ω1+xν11δ1)γ1β2(x2ω2+xν22δ2)γ2), (3.3)

    and

    f(x1,x2)=β1β2γ1γ2eβ1(x1ω1+xν11δ1)γ1β2(x2ω2+xν22δ2)γ2(x1ω1+xν11δ1)γ11×(x2ω2+xν22δ2)ω11(ω1+xν111δ1ν1)(ω2+xν122δ2ν2)×(1+ρ(12eβ1(x1ω1+xν11δ1)γ1β1γ1(x1ω1+xν11δ1)γ12(ω1+xν111δ1ν1))×(12eβ2(x2ω2+xν22δ2)γ2β2γ2(x2ω2+xν22δ2)γ12(ω2+xν122δ2ν2))). (3.4)

    Figure 2 presents the CDF for the bivariate GWMWD (0.1, 0.8, 4.4, 6.1, 3.20) and GWMWD (0.1, 0.8, 4.4, 6.1, 0.20) when the parameter ν increases, and for ρ=0.2.

    Figure 2.  CDF for the bivariate GWMWD (0.1, 0.8, 4.4, 6.1, 3.20) and GWMWD (0.1, 0.8, 4.4, 6.1, 0.20).

    The copula function is a way to construct bivariate distributions. Other methods can be reviewed and may be helpful to introduce some new bivariate distributions (see, Xu et al. [21] and Luo et al.[22].)

    Suppose that {x1,x2,...,xk} is an observed sample from XGWMWD(β,γ,ω,δ,ν). The likelihood function of β,γ,ω,δ, and ν becomes

    l=n!(nk)!eβ(nk)(Tω+Tνδ)γ(βγ)keβki=1(xiω+xνiδ)γki=1(ω+xν1iδν)(xiω+xνiδ)γ1, (4.1)

    and the log-likelihood function (L) is

    L=log[n!(nk)!]β(nk)(Tω+Tνδ)γ+klog[βγ]βki=1(xiω+xνiδ)γ+ki=1log[ω+xν1iδν]+(γ1)ki=1log[xiω+xνiδ]. (4.2)

    Let Q(x)=ωx+δxν. The first partial derivatives of (4.2) with respect to β,γ,ω,δ and ν are given by

    Lβ=kβ(nk)Q(T)γki=1Q(xi)γ, (4.3)
    Lγ=kγ(nk)βQ(T)γlogQ(T)+ki=1logQ(xi)βki=1Q(xi)γlogQ(xi), (4.4)
    Lω=(nk)TβγQ(T)γ1+ki=11ω+δνxν1i+(γ1)ki=1xiQ(xi)βγki=1xiQ(xi)γ1, (4.5)
    Lδ=(nk)TνβγQ(T)γ1+ki=1νxν1iω+δνxν1i+(γ1)ki=1xνiQ(xi)βγki=1xνiQ(xi)γ1, (4.6)
    Lν=(nk)TνβγδQ(T)γ1log[T]+δki=1xν1i+νlog[xi]xν1iω+δνxν1i+(γ1)δki=1log[xi]xνiQ(xi)βγδki=1log[xi]xνiQ(xi)γ1. (4.7)

    The maximum likelihood estimators ˆβML,ˆγML,ˆωML,ˆδML, and ˆνML of the GWMWD (β,γ,ω,δ,ν) parameters are the solutions of (4.3)–(4.7). The asymptotic confidence intervals of the parameters β,γ,ω,δ and ν can be calculated. ˆV=V(ˆβML,ˆγML,ˆωML,ˆδML,ˆνML) is the observed variance covariance matrix, such that

    V(β,γ,ω,δ,ν)=[2Lβ22Lβγ2Lβω2Lβδ2Lβν2Lγβ2Lγ22Lγω2Lγδ2Lγν2Lωβ2Lωγ2Lω22Lωδ2Lων2Lδβ2Lδγ2Lδω2Lδ22Lδν2Lνβ2Lνγ2Lνω2Lνδ2Lν2]1, (4.8)

    where

    2Lβ2=kβ2, (4.9)
    2Lβγ=(kn)Q(T)γlogδ2ki=1Q(xi)γlogQ(xi), (4.10)
    2Lβω=(kn)TγQ(T)γ1γki=1xiQ(xi)γ1, (4.11)
    2Lβδ=(kn)TνγQ(T)γ1γki=1xνiQ(xi)γ1, (4.12)
    2Lβν=(kn)TνγδQ(T)γ1log[T]γδki=1xνiQ(xi)γ1log[xi], (4.13)
    2Lγ2=kγ2(kn)βQ(T)γlogQ(T)2βki=1Q(xi)γlog[Q(xi)]2, (4.14)
    2Lγω=(kn)2T2β2γQ(T)2γ2logQ(T)+ki=1xiQ(xi)βki=1xiQ(xi)γ1(1+γlogQ(xi)), (4.15)
    2Lγδ=(kn)2T2νβ2γQ(T)2γ2logQ(T)+ki=1xνiQ(xi)βki=1xνiQ(xi)γ1(1+γlogQ(xi)), (4.16)
    2Lγν=(kn)TνβδQ(T)γ1log[T](1+γlogQ(T))+ki=1δlog[xi]xνiQ(xi)βδki=1xνiQ(xi)γ1log[xi](1+γlogQ(xi)), (4.17)
    2Lω2=(kn)T2β(γ1)γQ(T)γ2+ki=11Q(xi)2(γ1)ki=1x2iQ(xi)2+βγ(1γ)ki=1xi2Q(xi)γ2, (4.18)
    2Lωδ=(kn)T1+νβ(γ1)γQ(T)γ2+ki=1νxν1i(ω+δνxν1i)2(γ1)ki=1x1+νiQ(xi)2+βγki=1xiν+1Q(xi)γ2(1γ), (4.19)
    2Lων=(kn)T1+νβ(γ1)γδQ(T)γ2log[T](γ1)ki=1δlog[xi]x1+νiQ(xi)2δki=1(xν1i(ω+δνxν1i)2+νlog[xi]xν1i(ω+δνxν1i)2)+βγδki=1xiν+1Q(xi)γ2log[xi](1γ), (4.20)
    2Lδ2=(kn)T2νβ(γ1)γQ(T)γ2ki=1ν2x2+2νi(ω+δνxν1i)2ki=1(γ1)x2νiQ(xi)2+βγki=1x2νiQ(xi)γ2(1+γ), (4.21)
    2Lδν=(kn)TνβγQ(T)γ1log[T](Tν(γ1)δ(ωT+δTν)11)+(γ1)ki=1log[xi]xνiQ(xi)(1δ xνiQ(xi))+ki=1xν1iω+δνxν1i(1+νlog[xi]νxν1i(δ(1νlog[xi])ω+δνxν1i))βγki=1xνilog[xi](Q(xi)γ1+(γ1)δQ(xi)γ2), (4.22)
    2Lν2=(kn)2T3νβ2(γ1)γ2δ3Q(T)2γ3(log[T]2)2+(γ1)ki=1(δ2log[xi]2x2νiQ(xi)2+δlog[xi]2xνiQ(xi))+2δlog[xi]xν1i+δνlog[xi]2xν1iω+δνxν1iδ2ki=1x2ν2i(1+νlog[xi])(1(ω+νxν1i)2+νlog[xi](ω+δνxν1i)2)βγδki=1xνiQ(xi)γ1log[xi](log[xi]+(xνi)1(γ1)). (4.23)

    An approximate 100(1ϵ)% two-sided C.Is for the parameters β,γ,ω,δ and ν are

    ˆβ±zϵ/2V(ˆβ), (4.24)
    ˆγ±zϵ/2V(ˆγ), (4.25)
    ˆω±zϵ/2V(ˆω), (4.26)
    ˆδ±zϵ/2V(ˆδ), (4.27)

    and

    ˆν±zϵ/2V(ˆν), (4.28)

    respectively, where the diagonal elements of ˆV V(ˆβ), V(ˆγ), V(ˆω), V(ˆδ), and V(ˆν) are the estimated variances of ˆβML,ˆγML,ˆωML,ˆδML, and ˆνML, and zϵ/2 is the upper (ϵ2) percentile of the normal(0, 1) distribution.

    Let U have a uniform (0, 1) distribution. The GWMWD can be simulated by using the solution of the nonlinear equation

    0=1uexpβ(xω+xνδ)γ. (5.1)

    We simulate the GWMWD for two sets of the parameters: Set 1: β=1.4,γ=3.0,ω=0.7,δ=1.3,ν=0.4, and Set 2: β=0.4,γ=1.0,ω=1.7,δ=1.5,ν=1.8. The empirical results of the Monte Carlo simulation study are given in Table 1 for Set 1. The empirical results of the Monte Carlo simulation study are given in Table 2 for Set 2. Suppose that the data were observed for the GWMWD under the censoring scheme C-UHCS(m,r;T1,T2) and set the arbitrary values for termination as T=Xk and k=45n. The simulation study is carried out as follows

    Table 1.  Point and interval estimation of the parameters for β=1.4,γ=3.0,ω=0.7,δ=1.3, and ν=0.4 with different values of n.
    Par. n Bias MSEs 90%L 90%U 90%W 95%L 95%U 95%W
    ˆβ 25 1.284 1.955 0.001 5.89 5.89 0.001 6.516 6.516
    50 1.163 1.479 0.137 4.989 4.852 0.001 5.462 5.462
    100 1.124 1.325 0.35 4.698 4.347 0.001 5.122 5.122
    200 1.099 1.241 0.464 4.533 4.069 0.067 4.93 4.863
    400 1.08 1.18 0.545 4.415 3.87 0.167 4.793 4.625
    600 1.076 1.169 0.559 4.394 3.835 0.185 4.768 4.583
    800 1.079 1.171 0.559 4.399 3.841 0.184 4.774 4.59
    1000 1.078 1.169 0.561 4.395 3.834 0.187 4.77 4.582
    ˆγ 25 -0.307 0.403 2.032 3.354 1.322 1.903 3.483 1.58
    50 -0.428 0.311 2.061 3.082 1.021 1.962 3.182 1.22
    100 -0.475 0.288 2.053 2.997 0.944 1.961 3.089 1.129
    200 -0.502 0.281 2.038 2.959 0.921 1.948 3.049 1.101
    400 -0.51 0.276 2.038 2.941 0.904 1.949 3.03 1.08
    600 -0.52 0.28 2.02 2.939 0.919 1.931 3.029 1.099
    800 -0.521 0.279 2.021 2.937 0.915 1.932 3.026 1.094
    1000 -0.524 0.28 2.018 2.935 0.918 1.928 3.025 1.097
    ˆω 25 0.99 1.265 0.616 4.764 4.148 0.212 5.169 4.957
    50 0.886 0.921 1.076 4.096 3.02 0.781 4.39 3.609
    100 0.827 0.743 1.308 3.746 2.438 1.07 3.984 2.914
    200 0.798 0.668 1.403 3.593 2.19 1.189 3.807 2.617
    400 0.783 0.629 1.452 3.515 2.062 1.251 3.716 2.465
    600 0.78 0.619 1.466 3.495 2.029 1.268 3.693 2.425
    800 0.777 0.61 1.476 3.477 2.002 1.281 3.673 2.392
    1000 0.772 0.601 1.486 3.457 1.971 1.294 3.649 2.355
    ˆδ 25 1.102 1.355 0.379 4.824 4.445 -0.054 5.258 5.312
    50 1.057 1.212 0.57 4.544 3.974 0.182 4.932 4.75
    100 1.017 1.086 0.735 4.298 3.563 0.387 4.646 4.258
    200 1.011 1.052 0.786 4.236 3.449 0.45 4.572 4.123
    400 0.986 0.987 0.867 4.105 3.238 0.551 4.421 3.869
    600 0.978 0.966 0.893 4.063 3.17 0.584 4.373 3.788
    800 0.978 0.965 0.896 4.06 3.164 0.588 4.369 3.781
    1000 0.977 0.961 0.902 4.052 3.151 0.594 4.36 3.765
    ˆν 25 0.4 0.16 0.938 1.462 0.525 0.886 1.514 0.627
    50 0.4 0.16 0.938 1.462 0.525 0.886 1.514 0.627
    100 0.4 0.16 0.938 1.462 0.525 0.886 1.514 0.627
    200 0.4 0.16 0.938 1.462 0.525 0.886 1.514 0.627
    400 0.4 0.16 0.938 1.462 0.525 0.886 1.514 0.627
    600 0.4 0.16 0.938 1.462 0.525 0.886 1.514 0.627
    800 0.4 0.16 0.938 1.462 0.525 0.886 1.514 0.627
    1000 0.4 0.16 0.938 1.462 0.525 0.886 1.514 0.627

     | Show Table
    DownLoad: CSV
    Table 2.  Point and interval estimation of the parameters for β=0.4,γ=1.0,ω=1.7,δ=1.5, and ν=1.8 with different values of n.
    Par. n Bias MSEs 90%L 90%U 90%W 95%L 95%U 95%W
    ˆβ 25 0.498 0.303 0.401 1.394 0.994 0.304 1.491 1.187
    ˆβ 25 0.498 0.303 0.401 1.394 0.994 0.304 1.491 1.187
    50 0.458 0.226 0.487 1.229 0.742 0.415 1.302 0.887
    100 0.437 0.197 0.514 1.161 0.647 0.451 1.224 0.774
    200 0.426 0.187 0.519 1.133 0.613 0.459 1.193 0.733
    400 0.41 0.177 0.519 1.101 0.582 0.462 1.158 0.695
    600 0.404 0.175 0.517 1.091 0.574 0.461 1.147 0.686
    800 0.396 0.171 0.516 1.077 0.561 0.461 1.131 0.67
    1000 0.385 0.165 0.514 1.055 0.541 0.461 1.108 0.647
    ˆγ 25 -0.098 0.073 0.783 1.021 0.238 0.76 1.044 0.284
    50 -0.147 0.034 0.797 0.908 0.111 0.787 0.919 0.132
    100 -0.159 0.03 0.792 0.89 0.098 0.782 0.9 0.117
    200 -0.164 0.03 0.787 0.884 0.097 0.778 0.894 0.116
    400 -0.16 0.028 0.794 0.885 0.091 0.785 0.894 0.109
    600 -0.158 0.027 0.798 0.887 0.089 0.789 0.896 0.107
    800 -0.157 0.027 0.799 0.888 0.089 0.79 0.896 0.106
    1000 -0.154 0.027 0.802 0.89 0.088 0.794 0.898 0.105
    ˆω 25 -0.796 0.691 0.001 2.037 2.037 0.001 2.258 2.258
    50 -0.836 0.716 0.001 2.038 2.038 0.001 2.267 2.267
    100 -0.855 0.741 0.001 2.06 2.06 0.001 2.297 2.297
    200 -0.849 0.737 0.001 2.059 2.059 0.001 2.295 2.295
    400 -0.83 0.723 0.001 2.057 2.057 0.001 2.288 2.288
    600 -0.815 0.709 0.001 2.049 2.049 0.001 2.276 2.276
    800 -0.802 0.699 0.001 2.044 2.044 0.001 2.268 2.268
    1000 -0.783 0.683 0.001 2.037 2.037 0.001 2.255 2.255
    ˆδ 25 -0.597 0.415 0.222 1.584 1.362 0.089 1.717 1.628
    50 -0.637 0.422 0.17 1.556 1.386 0.035 1.691 1.656
    100 -0.654 0.436 0.131 1.561 1.43 0.001 1.701 1.709
    200 -0.652 0.436 0.134 1.563 1.429 0.001 1.702 1.707
    400 -0.657 0.441 0.12 1.565 1.445 0.001 1.706 1.706
    600 -0.651 0.435 0.136 1.563 1.427 0.001 1.702 1.702
    800 -0.638 0.428 0.161 1.564 1.403 0.024 1.701 1.677
    1000 -0.627 0.421 0.183 1.563 1.38 0.048 1.698 1.65
    ˆν 25 -0.899 0.863 0.001 2.316 2.316 0.001 2.592 2.592
    50 -0.941 0.904 0.001 2.341 2.341 0.001 2.63 2.63
    100 -0.953 0.921 0.001 2.356 2.356 0.001 2.651 2.651
    200 -0.948 0.918 0.001 2.358 2.358 0.001 2.652 2.652
    400 -0.924 0.897 0.001 2.347 2.347 0.001 2.635 2.635
    600 -0.905 0.876 0.001 2.332 2.332 0.001 2.612 2.612
    800 -0.894 2.33 0.001 2.848 2.848 0.001 2.6005 2.601
    1000 -0.874 2.319 0.001 2.785 2.785 0.001 2.59 2.59

     | Show Table
    DownLoad: CSV

    1) Random samples of size n=25,50,...,1000 were simulated from the GWMWD.

    2) The model parameters were estimated via the maximum likelihood method.

    3) 1000 iterations were made to obtain the MLEs, biases and MSEs of these estimators.

    4) Let ˆϑ be the MLE of ϑ=(β,γ,ω,δ,ν). The MLEs, biases and MSEs are given, respectively, by

    ˆϑ=11000Mi=1ˆϑi, (5.2)
    Bias(ˆϑ)=11000Mi=1(ˆϑiϑ), (5.3)

    and

    MSE(ˆϑ)=110001000i=1(ˆϑiϑ)2, (5.4)

    5) The 90% and 95% approximate confidence intervals with their width were calculated.

    This section is devoted for illustrating the GWMWD through the analysis of a reliability engineering application. The data set represents the failure times of 50 electronic components (per 1000 h); see Aryal and Elbatal [23]. Suppose that the data was observed from GWMWD under the censoring scheme C-UHCS(m,r;T1,T2), and set the arbitrary values for termination k=45 and T=10.943. Table 3 shows a summary of the reliability engineering data. The boxplot and Q-Q plot for the reliability engineering data are shown in Figure 4. The estimated parameters are ˆβ=0.058,ˆγ=9.986,ˆω=0.005,ˆδ=1.239, and ˆν=0.056. Plots of the fitted density and distribution functions of the GWMWD model are shown in Figure 5. The likelihood probability (PP) and Kaplan-Meier survival curve are shown in Figure 6.

    Figure 3.  Boxplot and Q-Q plot for the reliability engineering data.
    Table 3.  Reliability engineering data summary.
    Min. 1st Qu. Median Mean 3rd Qu. Max.
    0.058 0.254 1.600 3.410 4.534 15.080

     | Show Table
    DownLoad: CSV
    Figure 4.  Plots of fitted PDF and CDF of the GWMWD.
    Figure 5.  Plots of the PP and the Kaplan–Meier survival function of the GWMWD.

    Table 4 compares the GWMWD based on some detection criteria, such as the Akaike information criterion (AIC), Bayesian information criterion (BIC), Hannan-Quinn information criterion (HQIC) and consistent Akaike information criterion (CAIC). The goodness-of-fit results of the GWMWD model are compared with some other models, including the generalized Weibull distribution (GWWD), the generalized Weibull-Rayleigh distribution (GWRD), the generalized linear Weibull exponential distribution (GLWEXPD) and the generalized Weibull exponential distribution (GWEXPD). Table 5 compares the GWMWD with the Kolmogorov-Smirnov test for one sample. The results in Tables 4 and 5 suggest that the GWMWD provides a better fit than other competing models and could be chosen as a suitable model for analyzing heavy-tailed electronic data.

    Table 4.  Relative quality of the NG-MW vs competing models.
    Model AIC CAIC BIC HQIC
    GWMWD(0.058, 9.986, 0.005, 1.239, 0.056) 212.9574 214.3211 222.5175 216.5980
    GWWD(1.916, 0.972, 0.326, 0.635) 213.6829 214.5718 221.3310 216.5953
    GWRD(0.360, 0.246, 3.774) 221.4256 221.9473 227.1617 223.609
    GLWEXPD(0.466, 0.455, 0.715) 213.2008 213.7226 218.9369 215.3852
    GWEXPD(0.465, 0.640, 0.902, 0.007) 216.9577 217.8465 224.6058 219.8701

     | Show Table
    DownLoad: CSV
    Table 5.  One-sample Kolmogorov-Smirnov (KS) test.
    Model KS p-value
    GWMWD(0.058, 9.986, 0.005, 1.239, 0.056) 0.13263 0.3144
    GWWD(1.916, 0.972, 0.326, 0.635) 0.16936 0.2247
    GWRD(0.360, 0.246, 3.774) 0.14916 0.1952
    GLWEXPD(0.466, 0.455, 0.715) 0.14875 0.1977
    GWEXPD(0.465, 0.640, 0.902, 0.007) 0.17732 0.07589

     | Show Table
    DownLoad: CSV

    A new extension of the Weibull distribution i.e., the generalized modified Weibull distribution with five parameters, is presented. The model has a high degree of flexibility to fit the data appropriately. The provided model exhibits strong tail-heavy behavior and has unimodal increasing failure rate functions. Based on a combined-unified hybrid sample, the maximum likelihood estimators of the intended model parameters and a Monte Carlo simulation study were obtained. To illustrate the applicability and potential of the intended distribution, a dataset of failure times of 50 electronic components was analyzed. The mean square errors and biases decrease with increasing sample size. It is clear that the proposed model agrees well with the estimated PDF and CDF plots. The boxplot shows that the electronic downtime data set has a highly right skewed tail. The new generalized modified Weibull distribution based on the one-sample Kolmogorov-Smirnov test provides a better fit than other competing models. The proposed model fits the Kaplan-Meier survival plot very well. The results indicate that the generalized Weibull distribution (modified Weibull distribution) is considered ideal for modeling the intended engineering data. For future studies, we hope to discuss the accelerated life testing based on the new distribution by using stress-strength models (Zhang et al. [24,25]).

    The study was funded by Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R226), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

    The authors declare that they have no conflicts of interest.



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