Bayesian estimation of the continuous Weibull distribution parameters was studied by Ahmad and Ahmad (2013) under the assumption of knowing the shape parameter. Bayesian estimates are considered here of the parameters of the discrete Weibull Type I [DW(I)] distribution and are obtained under two different assumptions: when the shape parameter is known, and when both parameters are independent random variables. A Mathcad program is performed to simulate data from the DW(I) distribution considering different values of the parameters and different sample sizes, and to obtain Bayesian parameter estimates. The resulted estimates are compared to the ML and proportion estimates obtained by Khan et al. (1989)

Ashour, Samir Kamel

Muiftah, Mohamed Salem Abdelwahab

English

Digital Commons@Wayne State University

Journal of Modern Applied Statistical Methods Volume 18 Issue 2 Article 12 7-24-2020 Bayesian Estimation of the Parameters of Discrete Weibull Type (I) Distribution Samir Kamel Ashour Cairo University, Cairo, Egypt, ashoursamir@hotmail.com Mohamed Salem Abdelwahab Muiftah Cairo University, Cairo, Egypt, mohamedmuftah1961@gmail.com Follow this and additional works at: https://digitalcommons.wayne.edu/jmasm  Part of the Applied Statistics Commons, Social and Behavioral Sciences Commons, and the Statistical Theory Commons Recommended Citation Ashour, S. K., & Muiftah, M. S. A. (2019). Bayesian estimation of the parameters of discrete Weibull type (I) distribution. Journal of Modern Applied Statistical Methods, 18(2), eP2740. doi: 10.22237/jmasm/1604189160 This Regular Article is brought to you for free and open access by the Open Access Journals at DigitalCommons@WayneState. It has been accepted for inclusion in Journal of Modern Applied Statistical Methods by an authorized editor of DigitalCommons@WayneState. Journal of Modern Applied Statistical Methods November 2019, Vol. 18, No. 2, eP2740. doi: 10.22237/jmasm/1604189160  Copyright © 2020 JMASM, Inc. ISSN 1538 − 9472    doi: 10.22237/jmasm/1604189160 | Accepted: February 24, 2018; Published: July 24, 2020. Correspondence: Mohamed Salem Abdelwahab Muiftah, mohamedmuftah1961@gmail.com    2 Bayesian Estimation of the Parameters of Discrete Weibull Type (I) Distribution Samir Kamel Ashour Cairo University Cairo, Egypt Mohamed Salem Abdelwahab Muiftah Cairo University Cairo, Egypt   Bayesian estimation of the continuous Weibull distribution parameters was studied by Ahmad and Ahmad (2013) under the assumption of knowing the shape parameter. Bayesian estimates are considered here of the parameters of the discrete Weibull Type I [DW(I)] distribution and are obtained under two different assumptions: when the shape parameter is known, and when both parameters are independent random variables. A Mathcad program is performed to simulate data from the DW(I) distribution considering different values of the parameters and different sample sizes, and to obtain Bayesian parameter estimates. The resulted estimates are compared to the ML and proportion estimates obtained by Khan et al. (1989).  Keywords: Discrete Weibull distribution, Bayesian estimates, prior distribution, posterior distribution  Introduction The discrete version of the Weibull distribution DW(I) was introduced by Nakagawa and Osaki (1975) as follows: If T is a random variable following the continuous Weibull distribution with a shape parameter α > 0 and a scale parameter θ > 0 [T ~ W(α, θ)], then the probability density function (pdf), the cumulative distribution function (cdf), the survival function (sf), and the failure rate (fr) of T are, respectively:  ASHOUR & MUIFTAH 3  ( )( ) ( )( )( )11f e , 0F P 1 eShtTtTtTTt t tt T tt et t − −−−−= =  = −== (1)  Then, by inserting the sf given by (1) in the discretization formula   ( ) ( ) ( )P S S 1T TY y y y= = − + ,  the probability mass function (pmf) of the DW(I) distribution will be given by:   ( ) ( )1P e e , 0,1,yyY y y  − +−= = − =   By substituting e–θ = q, the pmf, sf, and fr of the DW(I) are, respectively:   ( ) ( )( )( ) ( )11P = , 0 1, 0,1,S eh , 1yyyy yYy yYii yY y P q q q yy qPq q yP +−+== = −   == == = − −  The first two moments of the DW(I) distribution are given by   ( ) ( ) ( )0 0E and E 2 Ey yy yY q Y yq Y  = == = +  .  Khan et al. (1989) compared the estimates of the parameters q and α obtained by method of moments and method of proportions. Kulasecara (1994) discussed the approximate ML estimation of the parameters under right censoring and compared his estimators with the proportion estimators introduced by Khan et al. The pmf of the DW(I) distribution with parameters q and α was also given by   ( ) ( )1P , 1,2,y yY y q q y −= = − =  (2)  BAYESIAN ESTIMATION OF DW(I) DISTRIBUTION'S PARAMETERS 4 where α > 0 and 0 < q < 1. Bayesian Estimation of the Parameters Estimating Assuming that α is Known Given a random sample Y1, Y2,…, Yn from the DW(I) distribution given by (2), then the likelihood function of this sample is   ( ) ( )11L | j jny yjq y q q−= = −   .  According to the Jeffreys' rule (Jeffreys, 1946), the prior distribution g(u) of a given parameter u is defined depending on its support as follows:  i. If 0 < u < 1, then ( )( )1g1uu u−; ii. If 0 < u < ∞, then ( )1g uu .  Thus, the prior distribution of q may be considered to be   ( )( )1g , 0 11q qq q  −.  Then, the posterior distribution of q is given by   ( ) ( ) ( )( )( )( )( )1111π | g L |, 0 11, 0 11j jj jn y yjn y yjq y q q yq qqq qq qqK q q−=−= −    − −  =  − (3)  ASHOUR & MUIFTAH 5 The normalized constant (K) can be obtained by integrating (3) w.r.t. q:   ( )( )1110 1j jn y yjq qK dqq q−=  −    =  −    .  Thus, the posterior distribution of q given y will be   ( )( )( )( )( )111110π | , 0 1111j jj jn y yjn y yjq qq y qq q q q dqq q−=−= −  =   − −  − (4)  Using squared error loss function, the Bayesian estimate and the Bayes' risk will be respectively, the mean and the variance of the posterior distribution (4). The mean and the variance of the posterior distribution of q given y are, respectively:   ( ) ( )( )( )11 110 0E | π |1j jn y yjq q qq y q q y dq dqK q q−=  −    =  =  −      (5)  and   ( ) ( ) ( )( )( )( )( )2221 12 21 11 10 0Var E E1 1j jj jn ny yy yj jq q qq q q q q qdq dqK q q K q q  − −= == −          − −              = −    − −              (6)  The mean given by (5) and the variance given by (6) need numerical techniques to be obtained. BAYESIAN ESTIMATION OF DW(I) DISTRIBUTION'S PARAMETERS 6 Estimating Assuming that q and α are Independent Random Variables According to the Jeffreys' rule and assuming that q and α are independent, the prior distributions of q and α may be considered to be   ( )( )( )1 21 1g and g1qq q =−.  Then, the joint prior distribution of q and α is given by   ( ) ( ) ( )( )1 21π , g g , 0 1, 01q q qq q   =   −.  The bivariate posterior distribution of q and α is then given by   ( ) ( ) ( )( )( )( )( )1111π , | π , L , |, 0 1, 01, 0 1, 01j jj jn y yjn y yjq y q q yq qqq qq qqC q q  −=−= −     − −  =   − (7)  The normalized constant (C) can be obtained by integrating (7) w.r.t. q and α:   ( )( )( )( )1 11 11 10 0 0 01 1j jj jn ny yy yj jq q q qC dqd d dqq q q q    − − = =   − −      = =− −    . (8)  The marginal posterior distribution of q given α and y is given by   ( )( )( )1110π | , , 0 11j jn y yjq qq y d qC q q −=  −    =   −    , ASHOUR & MUIFTAH 7 and the marginal posterior distribution of α given q and y is given by   ( )( )( )11120π | , , 01j jn y yjq qq y dqC q q −=  −    =  −    .  Using squared error loss function, the Bayesian estimate and the Bayes' risk will be the mean and the variance of the marginal posterior distribution respectively. The mean and the variance of the marginal posterior distributions of q given α and y are, respectively:   ( )( )( )1110 0E | ,1j jn y yjq qq y q d dqC q q −=   −     =   −         (9)  and   ( ) ( ) ( )( )( )( )22111 220 0Var E EE | ,1j jn y yjq q qq qq d dq q yC q q −== −      −     = −      −        (10)  The mean and the variance of the marginal posterior distributions of α given q and y are, respectively:   ( )( )( )1110 0E | ,1j jn y yjq qq y dq dC q q  −=   −     =   −         (11)  and  BAYESIAN ESTIMATION OF DW(I) DISTRIBUTION'S PARAMETERS 8  ( ) ( ) ( )( )( )( )22111 220 0Var E EE | ,1j jn y yjq qdq d q yC q q    −== −      −     = −      −        (12)  The means given by (9) and (11) and the variances given by (10) and (12) need numerical techniques to be obtained. Simulation A Mathcad program is used to simulate data from DW(I) distribution for some values of q and α, three of them were used by Khan, et al. (1989), namely, (q = 0.6, α = 1.5), (q = 0.8, α = 2.0), and (q = 0.5, α = 0.5), the Bayesian estimates of q and α are obtained for sample sizes (n = 20, 40, 50, 100) using 1000 replications, and compared to the ML and proportion estimates obtained by Khan et al. Results It is observed, in all studied cases, that the summation of the probabilities (the cdf) is not always the unity at a specific value of the random variable. Distribution I: (q = 0.6, α = 1.5) Table 1 represents the pmf of the DW(I) (1.5, 0.6) (approximated to 10 decimals), which shows that the cdf reaches the value 1 when Y = 13, that is, the probabilities of Y = 14, Y = 15,… and so on, are very small and may be ignored. The first two moments and the variance of the distribution are approximately given by   E(Y) = 1.926873844, E(Y2) = 4.667398731, and Var(Y) = 0.9545559203. Distribution II: (q = 0.8, α = 0.2) Table 2 represents the pmf of the DW(I) (0.2, 0.8) (approximated to 10 decimals), which shows that the value 1 of the cdf is unreachable for any finite Y. The first two moments and the variance of the distribution are approximately given by   E(Y) = 31941.19912, E(Y2) = 13115582590, and Var(Y) = 12095342390. ASHOUR & MUIFTAH 9 Table 1. The pmf of the DW(I) (1.5, 0.6)  y P(y) cdf 1 0.4 0.4 2 0.3642145624 0.7642145624 3 0.1654393283 0.9296538907 4 0.0535499493 0.9832038400 5 0.0134874738 0.9966913138 6 0.0027597743 0.9994510882 7 0.0004710537 0.9999221419 8 0.0000683052 0.9999904471 9 0.0000085294 0.9999989765 10 0.0000009270 0.9999999035 11 0.0000000884 0.9999999919 12 0.0000000075 0.9999999994 13 0.0000000006 1.0000000000 14 3.755306 × 10-10 1.0000000000 ⋮ ⋮ ⋮ 20 4.0898 × 10-19 1.0000000000 ⋮ ⋮ ⋮ 100 2.9542 × 10-219 1.0000000000 ⋮ ⋮ ⋮ 125 4.6784 × 10-307 1.0000000000   Table 2. The pmf of the DW(I) (0.2, 0.8)  y P(y) cdf 1 0.2 0.2 2 0.0261093004 0.2261093004 3 0.0165763282 0.2426856286 ⋮ ⋮ ⋮ 100 0.0006429482 0.4290828972 ⋮ ⋮ ⋮ 100000 0.0000004792 0.8926258176 ⋮ ⋮ ⋮ 1000000 0.0000000206 0.9708874799 ⋮ ⋮ ⋮ ∞ 0.0000000000 1.0000000000 Distribution III: (q = 0.5, α = 0.5) Table 3 represents the pmf of the DW(I) (0.5, 0.5) (approximated to 10 decimals), and shows that the cdf reaches the value 1 when Y = 955. The first two moments and the variance of the distribution are approximately given by   E(Y) = 4.788218708, E(Y2) = 108.6258571, and Var(Y) = 85.6988187. BAYESIAN ESTIMATION OF DW(I) DISTRIBUTION'S PARAMETERS 10 Table 3. The pmf of the DW(I) (0.5, 0.5)  y P(y) cdf 1 0.5 0.5 2 0.124785773 0.624785773 ⋮ ⋮ ⋮ 10 0.013298355 0.888298355 ⋮ ⋮ ⋮ 100 3.45264 × 10-5 0.999023438 ⋮ ⋮ ⋮ 955 5.61815 × 10-12 1.000000000 956 5.55259 × 10-12 1.000000000   Table 4. The pmf of the DW(I) (2.0, 0.8)  y P(y) cdf 1 0.2 0.2 2 0.3904 0.5904 3 0.2753822720 0.8657822720 4 0.1060702303 0.9718525023 5 0.0243696045 0.9962221068 6 0.0034533746 0.9996754814 ⋮ ⋮ ⋮ 10 0.0003066780 0.9999821594 11 0.0000172129 0.9999993723 12 0.0000006136 0.9999999859 13 0.0000000139 1.0000000000 14 0.0000000002 1.0000000000 ⋮ ⋮ ⋮ 57 1.2308 × 10-8 1.0000000000 58 0 1.00000000000 Distribution IV: (q = 0.8, α = 2.0) Table 4 represents the pmf of the DW(I) (2.0, 0.8) (approximated to 10 decimals), and shows that the cdf reaches the value 1 when Y = 13. The first two moments and the variance of the distribution are approximately given by   E(Y) = 2.376, E(Y2) = 6.687, and Var(Y) = 1.0416.  Table 5 represents the Bayesian estimates q̂B and r̂B of the scale parameter q and the Bayes' risk r of the suggested simulated DW(I) distributions for different sample sizes and their corresponding MSE's assuming that α is known.  ASHOUR & MUIFTAH 11 Table 5. Bayesian estimates of DW(I) parameters and their corresponding MSEs, Case 1: α is known  q α n q̂B MSE(q̂B) r̂B MSE(r̂B) × 10-7 0.6 1.5 20 0.478 0.024 0.006 9.830   40 0.488 0.017 0.003 18.320   50 0.487 0.017 0.003 17.390   100 0.491 0.014 0.001 33.990        0.5 0.5 20 0.461 0.009 0.006 1.986   40 0.474 0.005 0.003 0.707   50 0.471 0.004 0.003 0.140   100 0.475 0.002 0.001 1.427        0.8 0.2 20 0.606 0.039 0.004 1.251   40 0.609 0.037 0.002 0.310   50 0.608 0.037 0.002 0.203   100 0.609 0.037 0.001 0.525        0.8 2.0 20 0.703 0.015 0.003 8.799   40 0.717 0.009 0.002 3.003   50 0.715 0.009 0.002 12.540   100 0.722 0.007 0.001 238.300   Table 6. The Bayesian estimates of DW(I) parameters and their corresponding MSEs, Case 2: both α and q are unknown  q α n q̂B MSE(q̂B) ˆBα  MSE( ˆBα ) r̂B MSE(r̂B) × 10-6 0.6 1.5 20 0.431 0.039 3.534 84.959 0.010 2.060   40 0.430 0.035 1.726 2.833 0.006 13.480   50 0.425 0.035 1.577 0.299 0.005 28.530   100 0.424 0.034 1.468 0.143 0.001 913.400          0.5 0.5 20 0.446 0.013 0.268 0.074 0.010 46.592   40 0.449 0.008 0.268 0.081 0.005 9.573   50 0.445 0.007 0.221 0.084 0.004 6.324   100 0.441 0.005 0.200 0.091 0.002 1.383          0.8 0.2 20 0.883 0.013 0.442 0.182 0.003 21.977   40 0.901 0.017 0.405 0.112 0.002 11.333   50 0.896 0.015 0.373 0.073 0.003 6.943   100 0.909 0.084 0.347 0.064 0.001 2.248          0.8 2.0 20 0.612 0.046 2.973 8.326 0.010 19.620   40 0.616 0.040 2.525 0.893 0.006 32.740   50 0.613 0.040 2.464 0.727 0.004 53.580   100 0.614 0.037 2.337 0.341 0.001 887.200 BAYESIAN ESTIMATION OF DW(I) DISTRIBUTION'S PARAMETERS 12 Table 6 represents the Bayesian estimates q̂B, B̂ , and r̂B of the scale parameter q, shape parameter α, and the Bayes' risk r of the suggested simulated DW(I) distributions for different sample sizes and their corresponding MSE's assuming that q and α are independent random variables. From Tables 5 and 6 it may observed that the Bayesian estimates of the parameters q, α, and r become closer to the true parameter values as the sample size increases in most cases, which may implies the consistency of these estimates, the MSE's of the estimates become also smaller as the sample size increases in most cases. Tables 7 and 8 represent the estimates of the parameters q and α of the DW(I) distributions, respectively, and their corresponding variances obtained by Khan et al. (1989) accompanied with our Bayesian estimates. An inspection of Tables 7 and 8 shows that the estimates obtained by the method of moments seem to be more close to the true parameter values in most cases, it seems also that the variances of the Bayesian estimates of the scale parameter q are smaller in most cases than those of other estimates.   Table 7. The estimates of the parameter q of DW(I) and their corresponding variances     Method of moments  Method of proportions  Bayesian estimates      α is known  α is unkown q α n q̂M Var(q̂M)  q̂P Var(q̂P)  q̂B Var(q̂B)  q̂B Var(q̂B) 0.6 1.5 20 0.615 0.013  0.603 0.014  0.478 0.009  0.431 0.011   40 0.604 0.006  0.598 0.006  0.488 0.005  0.430 0.006   50 0.601 0.005  0.596 0.005  0.487 0.004  0.425 0.005   100 0.604 0.002  0.603 0.002  0.491 0.002  0.424 0.003               0.5 0.5 20 0.616 0.045  0.443 0.014  0.461 0.0008  0.453 0.010   40 0.590 0.032  0.437 0.007  0.474 0.0034  0.449 0.005   50 0.571 0.024  0.434 0.005  0.471 0.0029  0.445 0.006   100 0.551 0.013  0.432 0.003  0.475 0.0014  0.441 0.001               0.8 2.0 20 0.797 0.007  0.788 0.010  0.703 0.0058  0.612 0.010   40 0.796 0.004  0.791 0.005  0.717 0.0041  0.616 0.006   50 0.766 0.003  0.792 0.004  0.715 0.0020  0.613 0.005   100 0.801 0.001  0.800 0.002  0.722 0.0012  0.614 0.003         ASHOUR & MUIFTAH 13 Table 8. The estimates of the parameter α of DW(I) and their corresponding variances     Method of moments  Method of proportions  Bayesian estimates q α n ˆMα  Var( ˆMα )  ˆPα  Var( ˆPα )  ˆBα  Var( ˆBα ) 0.6 1.5 20 2.116 0.219  1.565 0.248  3.534 80.823   40 2.046 0.107  1.510 0.127  1.726 2.738   50 2.035 0.085  1.503 0.100  1.577 0.293   100 2.040 0.042  1.528 0.050  1.468 0.142            0.5 0.5 20 0.693 0.052  0.542 0.131  0.268 2.429   40 0.623 0.034  0.493 0.062  0.268 0.008   50 0.580 0.032  0.477 0.042  0.221 0.008   100 0.531 0.011  0.488 0.018  0.200 0.001            0.8 2.0 20 2.116 0.219  2.074 0.637  2.973 7.380   40 2.046 0.107  2.030 0.330  2.525 0.617   50 2.035 0.085  2.005 0.234  2.464 0.511   100 2.040 0.042  2.039 0.109  2.337 0.227 References Ahmad, S. P., & Ahmad, K. (2013). Bayesian analysis of Weibull distribution using R software. Australian Journal of Basic and Applied Sciences, 7(9), 156-164. Retrieved from http://www.ajbasweb.com/old/ajbas/2013/July/156-164.pdf Jeffreys, H. (1946). An invariant form for the prior probability in estimation problems. Proceedings of the Royal Society A, 186(1007), 453-461. doi: 10.1098/rspa.1946.0056 Khan, M. S. A., Khalique, A., & Aboummoh, A. M. (1989). On estimating parameters in a discrete Weibull distribution. IEEE Transactions on Reliability, 38(3), 348-350. doi: 10.1109/24.44179 Kulasecara, K. B. (1994). Approximate MLEs of the parameters of a discrete Weibull distribution with type-I censored data. Microelectronics Reliability, 34(7), 1185-1188. doi: 10.1016/0026-2714(94)90502-9 Nakagawa, T., & Osaki, S. (1975). The discrete Weibull distribution. IEEE Transactions on Reliability, R-24(5), 300-301. doi: 10.1109/TR.1975.5214915 

Bayesian Estimation of the Parameters of Discrete Weibull Type (I) Distribution

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Bayesian Estimation of the Parameters of Discrete Weibull Type (I) Distribution

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