On the Bayesian analysis of two-component mixture of transmuted Weibull distribution

Document Type : Article


1 Department of Mathematics and Statistics, Riphah International University, Islamabad, Pakistan

2 Department of Statistics, Quaid-i-Azam University, Islamabad, 45320, Pakistan


Transmuted distributions are skewed distributions and recently attracted a great attention of researchers due totheir applications in reliability and statistics. In this article, our main focus is on the Bayesian estimation of two-component mixture of Transmuted Weibull Distribution (TWD) under type-I right censored sampling scheme. In order to estimate the unknown parameters, non-informative and informative priors under Squared Error Loss Function (SELF), Precautionary Loss Function (PLF) and Quadratic Loss Function (QLF) are assumed when computing the posterior estimations. In addition the Bayesian credible intervals (BCI) were also constructed. Markov Chain Monte Carlo (MCMC) technique is adopted to generate samples from the posterior distributions and in turn computing different posterior summaries including Bayes estimates(BEs), posterior risks(PRs) and Bayesian credible intervals (BCI). As an illustration comparision of these Bayes estimators are made through simulated under different loss functions in terms of their respective posterior risks assuming different sample sizes and censoring rates. Two real-life examples; the first being the survival times of hepatitis B & C patientswhile the second being the hole diameter of 12 mm and the sheet thickness is 3.15 mm are also discussed to illustrate the potential application of the proposed methodology.


Main Subjects

1.    Shaw, W., and Buckley, I. The Alchemy of Probability Distributions: Beyond Gram-Charlier Expansions and a Skew-kurtotic-Normal Distribution from a Rank Transmutation Map (Research Report). King’s College, London, U.K, (2007).
2.    Bhati, D., Kumawat, P., and Gómez- D—źniz, E.“A New Count Model Generated from Mixed Poisson Transmuted Exponential Family with an Application to Health Care Data”,arXiv: 1504.01097v2 (2016).
3.    Shaw, W. T., and Buckley, I. R. C. “The Alchemy of Probability Distributions: Beyond Gram–Charlier Expansions and a Skew-Kurtotic Normal Distribution from a Rank Transmutation Map”. ArXiv Preprint: 0901.0434v1 [q-fin.ST] 5 Jan 2009, (2009).
4.    Jandhyala, V. K., Fotopoulos, S. B., and Evaggelopoulos , N. “Change-Point Methods for Weibull Models with Applications to Detection of Trends in Extreme Temperatures”, Environmetrics, 10, pp.547-564 (1999).
5.    Khan, M. S., King, R., and Hudson, I. L. “Transmuted New Generalized Weibull Distribution for Lifetime Modeling”, Communications for Statistical Applications and Methods, 23 (5), pp.363-383 (2016).
6.    Mudholkar, G. Srivastava, D. and Kollia, G. “A Generalization of the Weibull Distribution with Application to the Analysis of Survival Data”, Journal of the AmericanStatistical Association, 91(436), pp.1575–1583 (1996).
7.    Aryal, G. R., and Tsokos, C. P. “Transmuted Weibull distribution: A Generalization of the Weibull Probability Distribution”. European Journal of Pure and Applied Mathematics, 4 (2), pp. 89–102 (2011).
8.    Khan, M. S., and King, R. “Transmuted Modified Weibull Distribution: A Generalization of the Modified Weibull Probability Distribution”, European Journal of pure and applied mathematics, 6 (1), pp. 66-88 (2013).
9.    Merovci, F., Elbatal, I., and Ahmed, A. “The Transmuted Generalized Inverse Weibull Distribution”, Austrian Journal of Statistics ,43 (2), pp. 119-131 (2014).
10.    Abdurrahman, S. A. “Comparing Different Estimators of three Parameters for Transmuted Weibull Distribution”, Global Journal of Pure and Applied Mathematics,13(9), pp. 5115-5128 (2017).
11.    Nofal, Z. M., and El Gebaly, Y. M.“The Generalized Transmuted Weibull Distribution for Lifetime Data”, Pakistan Journal of Statistics and Operation Research, 8(2), pp. 355-378 (2017).
12.    Newcomb, S. “A Generalized Theory of the Combination of Observations so as to Obtain Best Result”, American Journal of Mathematics, 8, pp. 343-366 (1886).
13.    Feroze, N., and Aslam, M.“Bayesian Analysis of Doubly Censored Lifetime Data using Two-Component Mixture of Weibull Distribution”, .Journal of National Science Foundation of Sri Lanka, 42(4), pp. 325-334 (2014).
14.    Sindhu, T. N., Feroze, N., and Aslam, M. “Doubly Censored Data from Two - Component Mixture of Inverse Weibull Distributions: Theory and Applications”, Journal of Modern Applied Statistical Methods, 15(2), pp. 322-349 (2016).
15.    Aslam, M., Tahir, M., Hussain, Z., and Al-Zahrani, B.“A 3-Component Mixture of Rayleigh Distributions: Properties and Estimation in Bayesian Framework”, Public Library of Science One, pp. 1-14 (2015).
16.    Aslam, M., Tahir, M., and Hussain, Z. “Reliability Analysis of three-Component Mixture of Distributions”, Scientia Iranica, Transactions E: Industrial Engineering , 25 pp.1768-1781 (2018).
17.    Tahir, M., Aslam, M., Hussain. H., Abid, M., and Bhatti, S. H. “Bayesian Analysis of Heterogeneous Doubly Censored Lifetime Data using the 3-Component Mixture of Rayleigh Distributions: A Monte Carlo Simulation Study”, Scientia Iranic, 26 (3), pp. 1789-1808, (2019).
18.    Ateya, S. F.“Maximum Likelihood Estimation under a Finite Mixture of Generalized Exponential Distributions Based on Censored Data”, Statistical Papers, 55(2), pp. 311-325 (2014).
19.    Benaicha, H., and Chaker, A.“Weibull Mixture Modelfor Reliability Analysis”, International Review of Electrical Engineering, 9(5), pp. 986-990 (2014).
20.    Kalbfleisch, J. D. and Prentice, R. L. “The Statistical Analysis of Failure Time Data”, Wiley, New York, (1980).
21.    Gill, R. D. Censoring and Stochastic Integrals, Mathematical Centre Tracts 124. Mathematisch Centrum, Amsterdam, (1980).
22.    Miller, R. G. Jr. Survival Analysis. John Wiley and Sons, Inc., New York, pp.104-118 (1981).
23.    Mendenhall, W., and Hader, R. A. “Estimation of Parameters of Mixed Exponentially Distributed Failure Time Distributions from Censored Life Test Data”, Biometrika, 45(3-4), pp.504-520 (1958).
24.    Lindley, D.V. “Theory and Practice of Bayesian Statistics”, The Statistician, 32, pp.1-11 (1983).
25.    Walters, C., and Ludwig, D. “Calculation of Bayes Posterior Probability Distributions for Key Population Parameters: a Specified Approach”, Canadian Journal of Fisheries and Aquatic Sciences, 51(3), pp.713-722 (1994).
26.    Laplace, P. S.“Theorie Analytique Des Probabilities”. Veuve.Courcier, Paris, (1812).
27.    Punt, A. E., and Butterworth, D. S. “Why do Bayesian and Maximum Likelihood Assessments of the Bering–Chukchi– Beaufort Seas stock of bowhead whales differ”? Journal of Cetacean Research and Management, 2(2), pp.125–133 (2000).
28.    Punt, A. E., and Walker, T. I. “Stock Assessment and Risk Analysis for the School Shark Galeorhinus Galeus (Linnaeus) off Southern Australia”, Marina and Freshwater Research, 49, pp. 719–731 (1998).
29.    Abid, M., Naeem, A. , Hussain, Z., Riaz, M., and Tahir, M. “Investigating the Impact of Simple and Mixture Priors on Estimating Sensitive Proportion Through a General Class of Randomized Response Models”, Scientia Iranica, 26 (2), pp. 1009-1022, (2019).
30.    Franz, J.“Posterior Distribution and Loss Functions for Parameter Estimation in Weibull Processes”, Economic Quality Control, 21 (1), pp. 31 – 42 (2006).
31.    Ali, S., Aslam, M., Nasir, A., and Kazmi, S. M. A. “Scale Parameter Estimation of the Laplace Model Using Different Asymmetric Loss Functions”, International Journal of Statistics and Probability, 1(1), pp.105-127 (2012).
32.    Gauss, C. F.“Least Squares Method for the Combinations of Observations”, Translated by J.  Bertrand 1955, Mallet-Bachelier, Paris (1810).
33.    Legendre, A. M. Nouvelles Methods pour la determination des orbites des cometes F. Didot, (1805).  
34.    Norstrom, J. G. “The Use of Precautionary Loss Functions in Risk Analysis. IEEE Transactions on Reliability”, 45(1), pp. 400-403 (1996).
35.    Ali, S. “Mixture of the Inverse Rayleigh Distribution: Properties and Estimation in a Bayesian Framework”, Applied Mathematical Modelling, 39, pp. 515-530 (2015).
36.    Ali, S. “On the Bayesian Estimation of the Weighted Lindley Distribution”, Journal of Statistical Computation and Simulation,85 (5), pp.855–880 (2015).
37.    Casella, G., and George, E. I. “Explaining the Gibbs Sampler”, American. Statistician. 46(3), pp.167–174 (1992).
38.    Metropolis, N., and Ulam, S. “The Monte Carlo Method”, Journal of the American Statistical Association , 44,  pp. 335–341 (1949).
39.    Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A., and Teller, E. “Equations of State Calculation by Fast Computing Machines”, Journal of Chemical Physics, 21(6), pp.1087-1092 (1953).
40.    R Core Team R: A language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria.URL http://www.R-project.org/, (2013).
41.    Charnigo, R., Francoeur, M., Kenkel, P.,  Mengüc, M. P., Hall, B., and Srinivasan, C. “Credible Intervals for Nanoparticle Charactiristics”, Journal of Quantitative Spectroscopy & Radiative Transfer, 113(2), pp.182–193 (2012).
42.    Eberlya, L. E., & Casella, G. “Estimating Bayesian Credible Intervals”. Journal of Statistical Planning and Inference, 112, pp.115-132 (2003).
43.    Dasgupta, R. “On the distribution of Burr with applications”. Sankhya B, 73, pp. 1-19, (2011).
44.    Bakouch, H., Jamal, F., Chesneau, C., and Nasir, A. “A new transmuted family of distributions: Properties and estimation with applications”. https://hal.archives-ouvertes.fr/hal-01570370v3, (2017).