Research article Special Issues

Classical and Bayesian inference for the discrete Poisson Ramos-Louzada distribution with application to COVID-19 data


  • The present study is based on the derivation of a new extension of the Poisson distribution using the Ramos-Louzada distribution. Several statistical properties of the new distribution are derived including, factorial moments, moment-generating function, probability moments, skewness, kurtosis, and dispersion index. Some reliability properties are also derived. The model parameter is estimated using different classical estimation techniques. A comprehensive simulation study was used to identify the best estimation method. Bayesian estimation with a gamma prior is also utilized to estimate the parameter. Three examples were used to demonstrate the utility of the proposed model. These applications revealed that the PRL-based model outperforms certain existing competing one-parameter discrete models such as the discrete Rayleigh, Poisson, discrete inverted Topp-Leone, discrete Pareto and discrete Burr-Hatke distributions.

    Citation: Ibrahim Alkhairy. Classical and Bayesian inference for the discrete Poisson Ramos-Louzada distribution with application to COVID-19 data[J]. Mathematical Biosciences and Engineering, 2023, 20(8): 14061-14080. doi: 10.3934/mbe.2023628

    Related Papers:

    [1] Fathy H. Riad, Eslam Hussam, Ahmed M. Gemeay, Ramy A. Aldallal, Ahmed Z.Afify . Classical and Bayesian inference of the weighted-exponential distribution with an application to insurance data. Mathematical Biosciences and Engineering, 2022, 19(7): 6551-6581. doi: 10.3934/mbe.2022309
    [2] S. H. Sathish Indika, Norou Diawara, Hueiwang Anna Jeng, Bridget D. Giles, Dilini S. K. Gamage . Modeling the spread of COVID-19 in spatio-temporal context. Mathematical Biosciences and Engineering, 2023, 20(6): 10552-10569. doi: 10.3934/mbe.2023466
    [3] Kai Wang, Zhenzhen Lu, Xiaomeng Wang, Hui Li, Huling Li, Dandan Lin, Yongli Cai, Xing Feng, Yateng Song, Zhiwei Feng, Weidong Ji, Xiaoyan Wang, Yi Yin, Lei Wang, Zhihang Peng . Current trends and future prediction of novel coronavirus disease (COVID-19) epidemic in China: a dynamical modeling analysis. Mathematical Biosciences and Engineering, 2020, 17(4): 3052-3061. doi: 10.3934/mbe.2020173
    [4] Sarah R. Al-Dawsari, Khalaf S. Sultan . Modeling of daily confirmed Saudi COVID-19 cases using inverted exponential regression. Mathematical Biosciences and Engineering, 2021, 18(3): 2303-2330. doi: 10.3934/mbe.2021117
    [5] Manal M. Yousef, Rehab Alsultan, Said G. Nassr . Parametric inference on partially accelerated life testing for the inverted Kumaraswamy distribution based on Type-II progressive censoring data. Mathematical Biosciences and Engineering, 2023, 20(2): 1674-1694. doi: 10.3934/mbe.2023076
    [6] Mohamed S. Eliwa, Buthaynah T. Alhumaidan, Raghad N. Alqefari . A discrete mixed distribution: Statistical and reliability properties with applications to model COVID-19 data in various countries. Mathematical Biosciences and Engineering, 2023, 20(5): 7859-7881. doi: 10.3934/mbe.2023340
    [7] Francisco Julian Ariza-Hernandez, Juan Carlos Najera-Tinoco, Martin Patricio Arciga-Alejandre, Eduardo Castañeda-Saucedo, Jorge Sanchez-Ortiz . Bayesian inverse problem for a fractional diffusion model of cell migration. Mathematical Biosciences and Engineering, 2024, 21(4): 5826-5837. doi: 10.3934/mbe.2024257
    [8] Wael S. Abu El Azm, Ramy Aldallal, Hassan M. Aljohani, Said G. Nassr . Estimations of competing lifetime data from inverse Weibull distribution under adaptive progressively hybrid censored. Mathematical Biosciences and Engineering, 2022, 19(6): 6252-6275. doi: 10.3934/mbe.2022292
    [9] Walid Emam, Khalaf S. Sultan . Bayesian and maximum likelihood estimations of the Dagum parameters under combined-unified hybrid censoring. Mathematical Biosciences and Engineering, 2021, 18(3): 2930-2951. doi: 10.3934/mbe.2021148
    [10] Xiaomei Feng, Jing Chen, Kai Wang, Lei Wang, Fengqin Zhang, Zhen Jin, Lan Zou, Xia Wang . Phase-adjusted estimation of the COVID-19 outbreak in South Korea under multi-source data and adjustment measures: a modelling study. Mathematical Biosciences and Engineering, 2020, 17(4): 3637-3648. doi: 10.3934/mbe.2020205
  • The present study is based on the derivation of a new extension of the Poisson distribution using the Ramos-Louzada distribution. Several statistical properties of the new distribution are derived including, factorial moments, moment-generating function, probability moments, skewness, kurtosis, and dispersion index. Some reliability properties are also derived. The model parameter is estimated using different classical estimation techniques. A comprehensive simulation study was used to identify the best estimation method. Bayesian estimation with a gamma prior is also utilized to estimate the parameter. Three examples were used to demonstrate the utility of the proposed model. These applications revealed that the PRL-based model outperforms certain existing competing one-parameter discrete models such as the discrete Rayleigh, Poisson, discrete inverted Topp-Leone, discrete Pareto and discrete Burr-Hatke distributions.



    Data modeling has become extremely complicated in recent years as a result of the massive amount of data collected from many sectors, mainly in engineering, medicine, ecology, and renewable energy. The most popular option for analyzing count data sets is the Poisson distribution. The Poisson distribution has the drawback of being unable to represent overdispersed data sets. Overdispersion happens when the variation exceeds the mean. For count data sets, many researchers have presented mixed-Poisson distributions such as Poisson inverse Gaussian by [1], Conway–Maxwell–Poisson [2], Generalized Poisson Lindley [3], Poisson Weibull [4], Poisson Ishita [5], Poisson quasi-Lindley [6], Poisson Xgamma [7,8], Poisson XLindley [9], Poisson Moment Exponential [10], among authors. Even though there are several discrete models in the literature, there is still plenty of room to suggest a new discretized model that is acceptable under a variety of scenarios.

    Let X be a random variable having Ramos and Louzada distribution [11] with the probability density function (PDF) given by

    f(x;λ)=(τ22τ+x)τ2(τ1)e(xτ),    τ2,x>0. (1)

    where τ is the scale parameter.

    In this study, a new one-parameter discrete distribution for modeling count observations is introduced by compounding the Poisson distribution with Ramous-Louzada (RL) distribution. The resulting model is called the Poisson Ramous-Louzada (PRL) distribution. The major reason for the selection of the RL distribution as a compounding distribution is because of its simple form, which is needed to compute the statistical properties of the proposed distribution and estimate the unknown parameter. The proposed model may be used to model count datasets, which are frequently seen in real-world data modeling. To build a mixed Poisson model, it is assumed that the Poisson model's parameter is a random variable (RV) with a continuous distribution, and the count variable is drawn from the Poisson distribution conditional on the random parameter. As a result, the count variable's marginal distribution is a mixed Poisson distribution.

    The remainder of the paper is structured as follows: The new model is described in Section 2 and gives graphical representations of PMF, and HRF. Section 3 deduces several mathematical characteristics. Section 4 estimates the PRL parameter using the following classical estimation methods, maximum likelihood estimation (MLE), Anderson Darling (AD), Cramer von Mises (CVM), ordinary least-squares (OLS) and weighted least squares (WLS), and a simulation study is also given. Section 5 additionally discusses the Bayesian model formulation for the suggested distribution. Section 6 examines three real-world data sets to demonstrate the versatility of the PRL distribution. Section 6 also includes a Bayesian study of real-world data sets using Markov chain Monte Carlo methods. Section 7 concludes with some recommendations.

    A random variable X is said to follow the Poisson Ramos-Louzada distribution if it possesses the following stochastic representation

    (X|θ)Poisson(g(θ))
    (θ|τ)RL(τ)

    We call the marginal distribution of X the Poisson Ramos-Louzada distribution. The model is denoted by PRL(τ).

    Theorem 1: The PMF of PRL distribution is given by

    P(X=x,τ)=(1+1τ)x(x1+τ(τ1))(τ1)(1+τ)2;x=0,1,2,3,&τ2

    Proof: The PMF of the new probability model can be obtained as

    g(x|θ)=eθθxx!;x=0,1,2,3,&θ>0

    when its parameter θ follows RL distribution

    f(θ;τ)=(τ22τ+θ)τ2(τ1)e(θτ)

    We have

    P(X=x,τ)=0g(x|θ)f(θ;τ)dθ       =1x!τ2(τ1)0eθθx(τ22τ+θ)e(θτ)dθ       =1x!τ2(τ1)((τ22τ)0eθθxe(θτ)dθ+0eθθx+1e(θτ)dθ)       =1x!τ2(τ1)((τ22τ)(1+1τ)x1Γ(1+x)+(1+1τ)2xΓ(2+x))       P(X=x,τ)=(1+1τ)x(x1+τ(τ1))(τ1)(1+τ)2;x=0,1,2,3,&τ2. (2)

    The PMF behavior of the Poisson Ramos-Louzada distribution for various parameter values is shown in Figure 1.

    Figure 1.  PMF visualization plots for the PRL distribution.

    As can be seen, the PMF has a positively skewed and can be used to discuss the count data that is positively skewed. The corresponding CDF of the discrete Poisson Ramos-Louzada distribution is given as

    F(X=x)=pr(Xx)=1v=x+1P(v)            =1(1+1τ)xτ(x+τ2)(τ1)(1+τ)2;x=0,1,2,;τ2. (3)

    The corresponding survival function is

    S(x;τ)=(1+1τ)xτ(x+τ2)(τ1)(1+τ)2, (4)

    The hazard rate function (HRF), and reversed hazard rate function can be expressed as

    h(x;τ)=x+τ(τ1)1τ(x+τ2), (5)

    and

    r(x;τ)=1x+ττ2xτ+τ3(1+1τ)x(τ1)(1+τ)2. (6)

    The graphs below depict the behavior of the HRF of the discrete PRL distribution for various parameter values.

    Figure 2.  HRF visualization plots for the PRL distribution.

    This section has examined some statistical measures of the PRL distribution. Moments, the moment generating function (MGF), and the probability generation function are among them (pgf).

    Assume X is a PRL random variable, the rth factorial moments can be derived as

    μ(r)=E[E(X(r)|θ)],  where  X(r)=X(X1)(X2)(Xr+1)
    =1τ2(τ1)0[x=0x(r)eθθxx!](τ22τ+θ)e(θτ)dθ
    =1τ2(τ1)0[θrx=reθθxr(xr)!](τ22τ+θ)e(θτ)dθ

    Taking x+r in place of x within the bracket, we get

    μ(r)=1τ2(τ1)0[θrx=0eθθxx!](τ22τ+θ)e(θτ)dθ=1τ2(τ1)0θr(τ22τ+θ)e(θτ)dθ=τr(1+r+τ)Γ(1+r)τ1. (7)

    The first four factorial moments can be expressed as

    μ(1)=τ2τ1,
    μ(2)=2τ2(1+τ)τ1,
    μ(3)=6τ3(2+τ)τ1,

    and

    μ(4)=24τ4(3+τ)τ1.

    The first four moments about the mean of the PRL distribution are obtained.

    μ2=τ2(τ2+τ3)(τ1)2, (8)
    μ3=τ2(2τ4+3τ314τ2+4τ+7)(τ1)3, (9)
    μ4=τ2(9τ6+18τ592τ4+41τ3+77τ241τ15)(τ1)4, (10)

    Using Eqs (8)–(10), the Index of Dispersion (ID), coefficient of skewness (CS), and coefficient of Kurtosis (CK) can be derived in closed forms,

    ID(X)=Var(X)Mean(X)=τ2+τ3τ1, (11)
    CS(X)=μ3(μ2)32=τ2(7+4τ14τ2+3τ3+2τ4)(τ1)3(τ2(3+τ+τ2)(1+τ)2)3/2, (12)

    and

    CK(X)=9τ6+18τ592τ4+41τ3+77τ241τ15τ2(τ2+τ3)2. (13)

    The moment-generating function of RV X can be expressed as

    MX(s)=x=0exsP(X=x,τ)         =τ(τes(τ2)1)1(τ1)(1+τesτ)2. (14)

    The probability-generating function of PRL distribution can be derived as

    PX(t)=x=0txP(X=x,τ)       =1τ+2tτ+τ2tτ2(1+τ)(1τ+tτ)2. (15)

    Table 1 displays some computational statistics of the PRL distribution for sundry parameter values.

    Table 1.  Some computational statistics of PRL distribution.
    τ E(X) Var(X) CS(X) CK(X) ID(X) CV(X)
    2 4.00000 12.0000 1.44338 6.08333 3.00000 0.86603
    3 4.50000 20.2500 1.67901 7.05761 4.50000 1.00000
    4 5.33333 30.2222 1.79405 7.66025 5.66667 1.03078
    5 6.25000 42.1875 1.85607 8.02222 6.75000 1.03923
    6 7.20000 56.1600 1.89348 8.25493 7.80000 1.04083
    7 8.16667 72.1389 1.91786 8.41326 8.83333 1.04002
    8 9.14286 90.1224 1.93468 8.52586 9.85714 1.03833
    9 10.1250 110.1094 1.94678 8.60883 10.87500 1.03638
    10 11.1111 132.0988 1.95579 8.67174 11.88889 1.03441
    15 16.0714 272.0663 1.97871 8.83698 16.92857 1.02632
    20 21.0526 462.0499 1.98750 8.90278 21.94737 1.02103

     | Show Table
    DownLoad: CSV

    In this section, the parameter of PRL distribution is examined using some classical estimation approaches. The considered estimation approaches are maximum likelihood, Anderson-Darling, Cramer von Mises, least squares, and weighted least squares.

    Let X1,X2,X3,Xn be a random sample of failure times from PRL distribution, and the likelihood function for the parameter τ can be written as

    L(τ|x)=ni=1(1+1τ)xi(xi1+τ(τ1))(τ1)(1+τ)2, (16)

    and log-likelihood function is specified by

    l(τ|x)=ni=1log(1+1τ)xi+ni=1log(xi1+τ(τ1))nlog(τ1)nlog(1+τ)2. (17)

    We get the following equation by deriving Eq (17) with regard to parameter τ:

    lτ=ni=1xi(1+1τ)τ2+ni=12τ1xi+τ(τ1)1n(τ1)2n(τ+1). (18)

    The ML estimate is obtained by equating the above equation to zero and solving it for parameter τ. However, the ensuing expression has not a closed-form result and the required results can be obtained using iterative procedures.

    The Anderson-Darling (AD) estimator ˆτ of parameter τ can be defined by minimizing the following expression

    AD(τ)=n1nni=1(2i1)[log(F(x(i:n)|τ))+log(1F(x(i:n)|τ))],
    AD(τ)=n1nni=1(2i1)[log(1(1+1τ)x(i:n)τ(x(i:n)+τ2)(τ1)(1+τ)2)+log((1+1τ)x(i:n)τ(x(i:n)+τ2)(τ1)(1+τ)2)],

    Alternatively, the estimator can also be obtained by solving the following nonlinear equation

    ni=1(2i1)[ϕ(x(i:n)|τ)F(x(i:n)|τ)ϕ(x(n+1i:n)|τ)1F(x(n+1i:n)|τ)]=0

    where ϕ(xi:n|τ)=ddτF(x(i:n)|τ) and it reduces to

    ϕ(xi:n|τ)=(1+1τ)x(i:n)(x(i:n)2(τ1)(τ3)τ2x(i:n)(1+τ3τ2+τ3))(τ1)2(1+τ)3 (19)

    The ordinary least-square (OLS) estimator of the PRL model parameter can be obtained by minimizing

    LSE(τ)=ni=1[F(x(i:n)|τ)in+1]2,

    with respect to the parameter τ. Moreover, the LSE of τ is also obtained by solving

    mi=1[1i1+n(1+1τ)x(i:n)τ(x(i:n)+τ2)(τ1)(1+τ)2]ϕ(xi:n|τ)=0,

    The WLS estimate (WLSE) of τ, say ˆτ, can be determined by minimizing

    WLSE(τ)=ni=1(n+1)2(n+2)i(ni+1)[F(x(i:n)|τ)in+1]2,

    with respect to τ. The WLSE of τ can also be obtained by solving

    ni=1(1+n)2(2+n)i(ni+1)[1i1+n(1+1τ)x(i:n)τ(x(i:n)+τ2)(τ1)(1+τ)2]ϕ(xi:n|τ)=0,

    In which ϕ(xi:n|τ) is presented in (19).

    The Cramer von Mises (CVM) is a minimum distance-based estimator. The CVM of the PRL distribution can be obtained by minimizing

    CVM(τ)=112n+ni=1[log(F(x(i:n)|τ))2i12n]2,

    with respect to the parameter τ.

    The CVME of τ is also obtained by solving

    ni=1[12i12n(1+1τ)x(i:n)τ(x(i:n)+τ2)(τ1)(1+τ)2]ϕ(xi:n|τ)=0.

    In this section, we performed a simulation study to evaluate the accuracy of all considered estimators. In the simulation run, we generate 10,000 samples of size n = 10, 25, 50,100,200, and 300 from PRL distribution and then calculate the average estimates (AE), absolute bias (AB), mean relative error (MRE) and mean square error (MSE). For this purpose, we consider the six sets of values of parameter τ. The simulation results are presented in Tables 27.

    Table 2.  Parameter Estimates based on simulated samples for the parameter τ=2.1.
    Measures n MLE OLSE WLSE ADE CVME
    AE 10 2.4683 3.0734 2.7620 3.0486 2.7156
    25 2.3209 2.4309 2.1472 2.5131 2.3049
    50 2.2357 2.1799 2.1001 2.2501 2.1488
    100 2.1808 2.1098 2.1000 2.1307 2.1045
    200 2.1416 2.1002 2.1000 2.1013 2.1004
    300 2.1274 2.1000 2.1000 2.1002 2.1000
    AB 10 0.3683 0.9734 0.6620 0.9486 0.6156
    25 0.2209 0.3309 0.0472 0.4131 0.2049
    50 0.1357 0.0799 0.0001 0.1501 0.0488
    100 0.0808 0.0098 0.0000 0.0307 0.0045
    200 0.0416 0.0002 0.0000 0.0013 0.0004
    300 0.0274 0.0000 0.0000 0.0002 0.0000
    MRE 10 0.1754 0.4635 0.3152 0.4517 0.2931
    25 0.1052 0.1576 0.0225 0.1967 0.0976
    50 0.0646 0.0381 0.0001 0.0715 0.0233
    100 0.0385 0.0046 0.0000 0.0146 0.0021
    200 0.0198 0.0001 0.0000 0.0006 0.0002
    300 0.0130 0.0000 0.0000 0.0001 0.0000
    MSE 10 0.7743 3.3333 2.0870 3.4064 2.3300
    25 0.2951 0.8431 0.0912 1.1611 0.5520
    50 0.1391 0.1614 0.0001 0.3712 0.0985
    100 0.0627 0.0157 0.0000 0.0651 0.0068
    200 0.0271 0.0003 0.0000 0.0025 0.0006
    300 0.0169 0.0000 0.0000 0.0006 0.0000

     | Show Table
    DownLoad: CSV
    Table 3.  Parameter Estimates based on simulated samples for the parameter τ=3.0.
    Measures n MLE OLSE WLSE ADE CVME
    AE 10 3.1396 4.3862 4.3030 4.1871 4.2023
    25 3.0268 4.0219 3.9440 3.8787 3.9351
    50 2.9950 3.9277 3.9411 3.7879 3.8614
    100 2.9948 3.9041 4.0457 3.7762 3.8804
    200 2.9964 3.9231 4.2116 3.8175 3.9187
    300 2.9970 3.9378 4.3245 3.8419 3.9348
    AB 10 0.1396 1.3862 1.3030 1.1871 1.2023
    25 0.0268 1.0219 0.9440 0.8787 0.9351
    50 0.0050 0.9277 0.9411 0.7879 0.8614
    100 0.0052 0.9041 1.0457 0.7762 0.8804
    200 0.0036 0.9231 1.2116 0.8175 0.9187
    300 0.0030 0.9378 1.3245 0.8419 0.9348
    MRE 10 0.0465 0.5763 0.5672 0.5519 0.5599
    25 0.0089 0.4499 0.4713 0.4301 0.4473
    50 0.0017 0.3944 0.4427 0.3736 0.3907
    100 0.0017 0.3580 0.4347 0.3302 0.3570
    200 0.0012 0.3318 0.4495 0.3062 0.3321
    300 0.0010 0.3237 0.4659 0.2977 0.3217
    MSE 10 1.5854 5.3423 4.9727 4.6497 4.8640
    25 0.7594 2.7425 2.7410 2.4596 2.6652
    50 0.4343 1.9387 2.1702 1.7096 1.8726
    100 0.2391 1.4819 1.9558 1.2643 1.4561
    200 0.1248 1.1972 1.9742 1.0271 1.1933
    300 0.0832 1.0952 2.0686 0.9387 1.0861

     | Show Table
    DownLoad: CSV
    Table 4.  Parameter Estimates based on simulated samples for the parameter τ=4.0.
    Measures n MLE OLSE WLSE ADE CVME
    AE 10 3.9375 5.2163 5.2168 5.1234 5.0765
    25 3.9262 4.9401 4.9726 4.8907 4.8748
    50 3.9620 4.9088 5.0659 4.8573 4.8859
    100 3.9713 4.8963 5.1787 4.8401 4.8697
    200 3.9834 4.8858 5.3031 4.8439 4.8789
    300 3.9988 4.8862 5.3711 4.8506 4.8756
    AB 10 0.0625 1.2163 1.2168 1.1234 1.0765
    25 0.0738 0.9401 0.9726 0.8907 0.8748
    50 0.0380 0.9088 1.0659 0.8573 0.8859
    100 0.0287 0.8963 1.1787 0.8401 0.8697
    200 0.0166 0.8858 1.3031 0.8439 0.8789
    300 0.0012 0.8862 1.3711 0.8506 0.8756
    MRE 10 0.0156 0.4834 0.4807 0.4714 0.4854
    25 0.0185 0.3538 0.3585 0.3405 0.3536
    50 0.0095 0.2878 0.3167 0.2742 0.2889
    100 0.0072 0.2466 0.3054 0.2321 0.2424
    200 0.0042 0.2264 0.3260 0.2157 0.2256
    300 0.0003 0.2229 0.3428 0.2138 0.2205
    MSE 10 2.9043 6.2750 6.0663 5.8402 6.1414
    25 1.3856 3.1215 3.0986 2.8793 3.0677
    50 0.7547 1.9790 2.2677 1.8032 2.0032
    100 0.3717 1.3758 1.8782 1.2184 1.3353
    200 0.1822 1.0527 1.9039 0.9610 1.0504
    300 0.1202 0.9624 2.0149 0.8867 0.9453

     | Show Table
    DownLoad: CSV
    Table 5.  Parameter Estimates based on simulated samples for the parameter τ=5.0.
    Measures n MLE OLSE WLSE ADE CVME
    AE 10 4.8793 6.2159 6.2417 6.1934 6.0518
    25 4.9283 5.9406 6.0539 5.9254 5.8979
    50 4.9480 5.9061 6.0826 5.8767 5.8738
    100 4.9839 5.8739 6.1814 5.8623 5.8668
    200 4.9940 5.8508 6.2729 5.8430 5.8573
    300 4.9858 5.8588 6.3355 5.8436 5.8443
    AB 10 0.1207 1.2159 1.2417 1.1934 1.0518
    25 0.0717 0.9406 1.0539 0.9254 0.8979
    50 0.0520 0.9061 1.0826 0.8767 0.8738
    100 0.0161 0.8739 1.1814 0.8623 0.8668
    200 0.0060 0.8508 1.2729 0.8430 0.8573
    300 0.0142 0.8588 1.3355 0.8436 0.8443
    MRE 10 0.0241 0.4372 0.4343 0.4289 0.4335
    25 0.0143 0.2971 0.3010 0.2896 0.2975
    50 0.0104 0.2376 0.2514 0.2249 0.2334
    100 0.0032 0.1979 0.2415 0.1921 0.1968
    200 0.0012 0.1769 0.2549 0.1739 0.1775
    300 0.0028 0.1737 0.2671 0.1705 0.1714
    MSE 10 4.5278 8.0561 7.8298 7.6115 7.7488
    25 1.9727 3.5841 3.6415 3.3645 3.5746
    50 0.9619 2.1944 2.3526 1.9809 2.1332
    100 0.4770 1.4498 1.9140 1.3619 1.4272
    200 0.2355 1.0642 1.8717 1.0091 1.0642
    300 0.1600 0.9522 1.9495 0.9070 0.9279

     | Show Table
    DownLoad: CSV
    Table 6.  Parameter Estimates based on simulated samples for the parameter τ=7.0.
    Measures n MLE OLSE WLSE ADE CVME
    AE 10 6.8852 8.3751 8.3080 8.2719 8.0785
    25 6.9508 7.9654 8.0939 7.9661 7.9015
    50 6.9772 7.9105 8.0665 7.8996 7.8877
    100 6.9808 7.8671 8.1001 7.8491 7.8347
    200 6.9902 7.8377 8.2073 7.8376 7.8279
    300 6.9994 7.8330 8.2567 7.8323 7.8247
    AB 10 0.1148 1.3751 1.3080 1.2719 1.0785
    25 0.0492 0.9654 1.0939 0.9661 0.9015
    50 0.0228 0.9105 1.0665 0.8996 0.8877
    100 0.0192 0.8671 1.1001 0.8491 0.8347
    200 0.0098 0.8377 1.2073 0.8376 0.8279
    300 0.0006 0.8330 1.2567 0.8323 0.8247
    MRE 10 0.0164 0.3971 0.3834 0.3728 0.3801
    25 0.0070 0.2536 0.2487 0.2396 0.2494
    50 0.0033 0.1922 0.1930 0.1835 0.1908
    100 0.0027 0.1539 0.1695 0.1468 0.1502
    200 0.0014 0.1307 0.1741 0.1286 0.1298
    300 0.0001 0.1240 0.1797 0.1228 0.1229
    MSE 10 8.0853 13.221 12.275 11.590 11.818
    25 3.1995 5.2182 4.9697 4.6070 5.0214
    50 1.5333 2.9054 2.8651 2.6487 2.8488
    100 0.7477 1.7950 2.0349 1.6308 1.7225
    200 0.3819 1.2139 1.8654 1.1544 1.1984
    300 0.2538 1.0264 1.8495 0.9997 1.0176

     | Show Table
    DownLoad: CSV
    Table 7.  Parameter Estimates based on simulated samples for the parameter τ=15.0.
    Measures n MLE OLSE WLSE ADE CVME
    AE 10 14.870 16.725 16.755 16.639 16.487
    25 14.975 16.192 16.194 16.166 16.071
    50 15.003 15.979 16.045 16.015 15.948
    100 15.014 15.881 16.023 15.876 15.818
    200 14.992 15.818 16.066 15.853 15.801
    300 15.014 15.810 16.081 15.826 15.813
    AB 10 0.1302 1.7254 1.7551 1.6394 1.4871
    25 0.0254 1.1917 1.1941 1.1658 1.0705
    50 0.0026 0.9791 1.0445 1.0145 0.9477
    100 0.0142 0.8806 1.0226 0.8755 0.8180
    200 0.0084 0.8184 1.0661 0.8532 0.8008
    300 0.0135 0.8103 1.0806 0.8263 0.8125
    MRE 10 0.0087 0.3328 0.3258 0.3120 0.3262
    25 0.0017 0.2138 0.2007 0.1975 0.2060
    50 0.0002 0.1496 0.1444 0.1445 0.1491
    100 0.0009 0.1114 0.1076 0.1043 0.1085
    200 0.0006 0.0832 0.0879 0.0814 0.0824
    300 0.0009 0.0733 0.0817 0.0710 0.0729
    MSE 10 27.483 42.816 40.605 37.786 40.567
    25 11.128 16.789 14.977 14.517 15.753
    50 5.4177 8.2091 7.5370 7.6602 8.2052
    100 2.7917 4.5139 4.1585 3.9290 4.2486
    200 1.3835 2.4799 2.6592 2.3612 2.4262
    300 0.8964 1.8851 2.2170 1.7610 1.8652

     | Show Table
    DownLoad: CSV

    The Bayesian parameter estimation technique is an alternate to classical maximum likelihood estimation. In Bayesian estimation, a prior distribution must be defined for each unknown parameter. Consider a set of data x=x1,x2,,xn taken from discrete PRL distribution and the likelihood function is provided by

    L(τ|x)=ni=1(1+1τ)xi(xi1+τ(τ1))(τ1)(1+τ)2. (20)

    The Bayesian model is constructed by stating the prior distribution for the model parameter and then multiplying it with the likelihood function for the provided data using the Bayes theorem to generate the posterior distribution function. The prior distribution of parameter τ is denoted as p(τ).

    p(τ|x)L(τ|x)p(τ).

    For the proposed distribution, the gamma distribution is considered a prior distribution with known hyperparameters such as τGamma(α,β). The posterior expression, up to proportionality, may be found by multiplying the likelihood by the prior, and this can be represented as

    p(τ|x)βαΓ(α)τα1exp(τβ)ni=1(1+1τ)xi(xi1+τ(τ1))(τ1)(1+τ)2

    The posterior density is not mathematically tractable; for inference purposes, we will utilize the Markov Chain Monte Carlo (MCMC) approach to mimic posterior samples, allowing for easy sample-based conclusions.

    In the present study, we explore the application of MCMC algorithms implemented in the package MCMCpack of the R program to simulate samples from the joint posterior distribution. For this purpose, we generated 1006000 samples of the joint posterior distribution of interest. The effects of the initial values in the iterative process are eliminated after a burn-in phase of 6000 simulated samples. To achieve approximately independent samples, a thinning interval of size 300 was utilized. The parameter Bayes estimates were gained by taking the expected value of generated samples. Traceplots and the Geweke diagnostic were used to monitor the convergence of the simulated sequences. The asymptotic standard error of the difference divided by the difference between the two means of non-overlapping parts of a simulated Markov chain is the basis of the Geweke convergence diagnostic. We may say that a chain has reached convergence if its corresponding absolute z score is smaller than 1.96 since this z score asymptotically follows a typical normal distribution. The construction of interesting posterior summaries was done using the R software package MCMCpack.

    This section is ardent to prove the usefulness of the discrete Poisson Ramos-Louzada distribution in the modeling of three datasets. We compare the fits of the proposed distribution with some renowned one-parameter discrete distributions, discrete Raleigh [12], Poisson, discrete Pareto [13] and discrete Burr-Hatke [14], discrete Inverted Topp-Leone [15]. The Kolmogorov-Smirnov (KS) test, Akaike Information Criteria (AIC) and Bayesian Information Criteria (BIC) are used to compare the fitted models. We also illustrate the estimation procedures based on censored samples proposed in the previous section with three examples from the literature.

    A sample of the failure time of 15 electronic components in an acceleration life test [16]. The observations are 1, 5, 6, 11, 12, 19, 20, 22, 23, 31, 37, 46, 54, 60, and 66. The mean and variance of the first dataset are 27.533 and 431.94 respectively. The dispersion index value is 15.689 which indicates that the dataset is overdispersed. We determine the MLEs, standard errors (SE), and model selection measures (AIC, BIC, and KS) for the first dataset using the R software's maxLik package. These results are shown in Table 8 along with the model selection measures.

    Table 8.  ML Estimates and goodness-of-fit for the first dataset.
    Model MLEs (S.E.) -LogLik. AIC BIC K-S P-value
    PRL 26.455 (7.2429) 64.995 131.99 132.70 0.1770 0.6700
    DR 24.382 (3.1481) 66.394 134.79 135.50 0.2160 0.4300
    Poisson 27.533 (1.3548) 151.21 304.41 305.12 0.3810 0.0180
    DITL 0.4178 (0.1079) 74.491 150.98 151.69 0.3590 0.0310
    DPr 0.3284 (0.0848) 77.402 156.80 157.51 0.4060 0.0097
    DBH 0.9992 (0.0076) 91.368 184.74 185.44 0.7910 0.0000

     | Show Table
    DownLoad: CSV
    Figure 3.  Plots of fitted CDFs versus empirical CDFs for the first dataset.

    For Bayesian data analysis, the parameter τ of the PRL distribution was assumed to have an approximate gamma as the prior distribution, that is, τGamma(0.001,0.1). Figure 4 depicts posterior samples for the parameter τ. The evaluation of the MCMC draws across iterations is assessed using traceplot, posterior density, and ACF plot. From the traceplot, it is interesting to note that the samples produced attained acceptable convergence. The ACF plot indicates that the posterior samples are uncorrelated. Furthermore, the z-score of the Geweke test is –0.2498, indicating that the samples have sufficiently converged to a stable distribution. The posterior mean for τ is τBayes=13.00418 with a standard deviation of 2.18641, and the corresponding 95% highest density interval is (9.008356, 17.3976). We observe that the ML and Bayesian estimates are quite similar.

    Figure 4.  Traceplot, Posterior density, and ACF plot based on the first dataset.

    A sample of 66 patients died due to COVID-19 in China from January 23, 2022, to March 28, 2020. The data are: 8, 16, 15, 24, 26, 26, 38, 43, 46, 45, 57, 64, 65, 73, 73, 86, 89, 97,108, 97,146,121,143,142,105, 98,136,114,118,109, 97,150, 71, 52, 29, 44, 47, 35, 42, 31, 38, 31, 30, 28, 27, 22, 17, 22, 11, 7, 13, 10, 14, 13, 11, 8, 3, 7, 6, 9, 7, 4, 6, 5, 3 and 5. Some descriptive measures (mean, variance, and dispersion index) for this dataset are 47.742, 1924.8, and 38.696. We acquire the ML estimates for the parameter, and model selection metrics (AIC, BIC, and KS) for the second dataset. These results are shown in Table 9.

    Table 9.  ML Estimates and goodness-of-fit for the second dataset.
    Model MLEs (S.E.) -LogLik. AIC BIC K-S P-value
    PRL 48.711 (6.1847) 324.51 651.02 653.21 0.0851 0.7300
    DR 47.010 (2.8934) 347.23 696.45 698.64 0.2930 0.0000
    Poisson 49.743 (0.8682) 1409.8 2821.6 2823.8 0.4970 0.0000
    DITL 0.3539 (0.0436) 366.91 735.81 738.00 0.3290 0.0000
    DPr 0.2863 (0.0352) 379.07 760.14 762.33 0.3820 0.0000
    DBH 0.9997 (0.0019) 461.02 924.04 926.23 0.8120 0.0000

     | Show Table
    DownLoad: CSV

    For Bayesian data analysis, the parameter tau of the PRL distribution was assumed to have a gamma prior distribution. The associated Geweke z-score is –0.08203, which likewise indicates that the samples have sufficiently converged to a stable distribution. The posterior mean for τ is τBayes=32.0684 with a standard deviation of 2.89397, and a 95% HDI of (26.20931, 37.44432). The ML and Bayesian estimates are discernibly similar to one another.

    Figure 5.  Plots of fitted CDFs versus empirical CDFs for the second dataset.
    Figure 6.  Traceplot, Posterior density, and ACF plot based on the second dataset.

    The third dataset is also about deaths due to COVID-19 in Pakistan from 18 March 2020 to 30 June 2020. The data are: 1, 6, 6, 4, 4, 4, 1, 20, 5, 2, 3, 15, 17, 7, 8, 25, 8, 25, 11, 25, 16, 16, 12, 11, 20, 31, 42, 32, 23, 17, 19, 38, 50, 21, 14, 37, 23, 47, 31, 24, 9, 64, 39, 30, 36, 46, 32, 50, 34, 32, 34, 30, 28, 35, 57, 78, 88, 60, 78, 67, 82, 68, 97, 67, 65,105, 83,101,107, 88,178,110,136,118,136,153,119, 89,105, 60,148, 59, 73, 83, 49,137 and 91. Some computational measures, mean, variance and index of dispersion for the third dataset are; 50.057, 1758.8, and 35.135. The MLEs and goodness-of-fit measures for this dataset are given in Table 10.

    Table 10.  ML Estimates and goodness-of-fit for the third dataset.
    Model MLEs (S.E.) -LogLik. AIC BIC K-S P-value
    PRL 49.020 (5.4201) 428.30 858.61 861.07 0.0676 0.8210
    DR 46.339 (2.4841) 452.55 907.10 909.56 0.2473 0.0000
    Poisson 50.058 (0.9742) 1713.0 3428.1 3430.5 0.4954 0.0000
    DITL 0.3493 (0.0375) 488.14 978.28 980.75 0.3263 0.0000
    DPr 0.2835 (0.0304) 503.61 1009.2 1011.7 0.3558 0.0000
    DBH 0.9997 (0.0016) 613.80 1229.6 1232.1 0.7876 0.0000

     | Show Table
    DownLoad: CSV
    Figure 7.  Plots of fitted CDFs versus empirical CDFs for the third dataset.

    For the third dataset, the gamma distribution is again considered as the prior distribution, and the posterior samples for the parameter are described in Figure 8. Furthermore, the Geweke z-score is used as a diagnostic measure and its value is –0.03794, suggesting convergence of the samples to a stable distribution. The posterior mean for the third dataset is τBayes=46.96159 with a standard deviation of 4.92385. The corresponding 95% HDI (37.94273, 57.07319). The ML and Bayes estimate is quite similar to each other.

    Figure 8.  Traceplot, Posterior density, and ACF plot based on the third dataset.

    In this paper, we introduce a one-parameter discrete distribution by compounding Poisson with the Ramos-Louzada distribution. The proposed distribution is showing unimodal and positively skewed behavior. The failure rate of new distribution is increasing pattern. Some statistical properties derived include the moment-generating function, probability-generating function, factorial moments, dispersion index, skewness and kurtosis. The model parameter is estimated using the maximum likelihood estimation approach and the behavior of the derived estimator is assessed via a simulation study. The usefulness of the proposed distribution is carried out using three real-life datasets. The proposed distribution provides more efficient results than all considered competitive distributions. The Bayesian analysis is also performed by taking the MCMC approximation approach.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors declare there is no conflict of interest.



    [1] M. Shoukri, M. H. Asyali, R. VanDorp, D. Kelton, The Poisson inverse Gaussian regression model in the analysis of clustered counts data, J. Data Sci., 2 (2004), 17–32. https://doi.org/10.6339/JDS.2004.02(1).135 doi: 10.6339/JDS.2004.02(1).135
    [2] G. Shmueli, T. P. Minka, J. B. Kadane, S. Borle, P. Boatwright, A useful distribution for fitting discrete data: revival of the Conway–Maxwell–Poisson distribution, J. R. Stat. Soc. Ser. C., 54 (2005), 127–142. https://doi.org/10.1111/j.1467-9876.2005.00474.x doi: 10.1111/j.1467-9876.2005.00474.x
    [3] E. Mahmoudi, H. Zakerzadeh, Generalized poisson–lindley distribution, Commun. Stat. Methods, 39 (2010), 1785–1798. https://doi.org/10.1080/03610920902898514 doi: 10.1080/03610920902898514
    [4] L. Cheng, S. R. Geedipally, D. Lord, The Poisson–Weibull generalized linear model for analyzing motor vehicle crash data, Saf. Sci., 54 (2013), 38–42. https://doi.org/10.1016/j.ssci.2012.11.002 doi: 10.1016/j.ssci.2012.11.002
    [5] H. Hassan, S. A. Dar, P. B. Ahmad, Poisson Ishita distribution: A new compounding probability model, IOSR J. Eng., 9 (2019), 38–46.
    [6] E. Altun, A new model for over-dispersed count data: Poisson quasi-Lindley regression model, Math. Sci., 13 (2019), 241–247. https://doi.org/10.1007/s40096-019-0293-5 doi: 10.1007/s40096-019-0293-5
    [7] B. A. Para, T. R. Jan, H. S. Bakouch, Poisson Xgamma distribution: A discrete model for count data analysis, Model Assist. Stat. Appl., 15 (2020), 139–151. https://doi.org/10.3233/MAS-200484 doi: 10.3233/MAS-200484
    [8] E. Altun, G. M. Cordeiro, M. M. Ristić, An one-parameter compounding discrete distribution, J. Appl. Stat., 49 (2022), 1935–1956. https://doi.org/10.1080/02664763.2021.1884846 doi: 10.1080/02664763.2021.1884846
    [9] M. Ahsan-ul-Haq, A. Al-bossly, M, El-morshedy, M. S. Eliwa, Poisson XLindley distribution for count data : Statistical and reliability properties with estimation techniques and inference, Comput. Intell. Neurosci., 2022 (2022). https://doi.org/10.1155/2022/6503670 doi: 10.1155/2022/6503670
    [10] M. Ahsan-ul-Haq, On poisson moment exponential distribution with applications, Ann. Data Sci., 2022. https://doi.org/10.1007/s40745-022-00400-0 doi: 10.1007/s40745-022-00400-0
    [11] P. L. Ramos, F. Louzada, A Distribution for instantaneous failures, Stats, 2 (2019), 247–258. https://doi.org/10.3390/stats2020019 doi: 10.3390/stats2020019
    [12] D. Roy, Discrete rayleigh distribution, IEEE Trans. Reliab., 53 (2004), 255–260. https://doi.org/10.1109/TR.2004.829161 doi: 10.1109/TR.2004.829161
    [13] H. Krishna, P. S. Pundir, Discrete Burr and discrete Pareto distributions, Stat. Methodol., 6 (2009), 177–188. https://doi.org/10.1016/j.stamet.2008.07.001 doi: 10.1016/j.stamet.2008.07.001
    [14] M. El-Morshedy, M. S. Eliwa, E. Altun, Discrete Burr-Hatke distribution with properties, estimation methods and regression model, IEEE Access, 8 (2020), 74359–74370. https://doi.org/10.1109/ACCESS.2020.2988431 doi: 10.1109/ACCESS.2020.2988431
    [15] A. S. Eldeeb, M. Ahsan-ul-Haq, A. Babar, A discrete analog of inverted Topp-Leone distribution: Properties, estimation and applications. Int. J. Anal. Appl., 19 (2021), 695–708. https://doi.org/10.28924/2291-8639-19-2021-695 doi: 10.28924/2291-8639-19-2021-695
    [16] J. F. Lawless, Statistical models and methods for lifetime data, John Wiley & Sons, 2011.
  • This article has been cited by:

    1. Fatimah M. Alghamdi, Muhammad Ahsan-ul-Haq, Muhammad Nasir Saddam Hussain, Eslam Hussam, Ehab M. Almetwally, Hassan M. Aljohani, Manahil SidAhmed Mustafa, Etaf Alshawarbeh, M. Yusuf, Discrete Poisson Quasi-XLindley distribution with mathematical properties, regression model, and data analysis, 2024, 17, 16878507, 100874, 10.1016/j.jrras.2024.100874
    2. Osama Abdulaziz Alamri, Classical and Bayesian estimation of discrete poisson Agu-Eghwerido distribution with applications, 2024, 109, 11100168, 768, 10.1016/j.aej.2024.09.063
    3. Amani Alrumayh, Marco Costa, Bernoulli Poisson Moment Exponential Distribution: Mathematical Properties, Regression Model, and Applications, 2024, 2024, 0161-1712, 10.1155/2024/5687958
    4. Seth Borbye, Suleman Nasiru, Kingsley Kuwubasamni Ajongba, Vladimir Mityushev, Poisson XRani Distribution: An Alternative Discrete Distribution for Overdispersed Count Data, 2024, 2024, 0161-1712, 10.1155/2024/5554949
    5. Waheed Babatunde Yahya, Muhammad Adamu Umar, A new poisson-exponential-gamma distribution for modelling count data with applications, 2024, 0033-5177, 10.1007/s11135-024-01894-x
    6. Yingying Qi, Dan Ding, Yusra A. Tashkandy, M.E. Bakr, M.M. Abd El-Raouf, Anoop Kumar, A novel probabilistic model with properties: Its implementation to the vocal music and reliability products, 2024, 107, 11100168, 254, 10.1016/j.aej.2024.07.035
    7. Abdullah Ali H. Ahmadini, Muhammad Ahsan-ul-Haq, Muhammad Nasir Saddam Hussain, A new two-parameter over-dispersed discrete distribution with mathematical properties, estimation, regression model and applications, 2024, 10, 24058440, e36764, 10.1016/j.heliyon.2024.e36764
    8. Safar M. Alghamdi, Muhammad Ahsan-ul-Haq, Olayan Albalawi, Majdah Mohammed Badr, Eslam Hussam, H.E. Semary, M.A. Abdelkawy, Binomial Poisson Ailamujia model with statistical properties and application, 2024, 17, 16878507, 101096, 10.1016/j.jrras.2024.101096
    9. Khlood Al-Harbi, Aisha Fayomi, Hanan Baaqeel, Amany Alsuraihi, A Novel Discrete Linear-Exponential Distribution for Modeling Physical and Medical Data, 2024, 16, 2073-8994, 1123, 10.3390/sym16091123
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1491) PDF downloads(78) Cited by(9)

Figures and Tables

Figures(8)  /  Tables(10)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog