We consider the estimation problem of the probability for Lomax distribution based on general progressive censored data. The maximum likelihood estimator and Bayes estimators are obtained using the symmetric and asymmetric balanced loss functions. The Markov chain Monte Carlo (MCMC) methods are used to accomplish some complex calculations. Comparisons are made between Bayesian and maximum likelihood estimators via Monte Carlo simulation study. 1. Introduction The Lomax distribution, also called “Pareto type II” distribution is a particular case of the generalized Pareto distribution (GPD). The Lomax distribution has been used in the literature in a number of ways. For example, it has been extensively used for reliability modelling and life testing; see, for example, Balkema and de Haan [1]. It also has been used as an alternative to the exponential distribution when the data are heavy tailed; see Bryson [2]. Ahsanullah [3] studied the record values of Lomax distribution. Balakrishnan and Ahsanullah [4] introduced some recurrence relations between the moments of record values from Lomax distribution. The order statistics from nonidentical right-truncated Lomax random variables have been studied by Childs et al. [5]. Also, the Lomax model has been studied, from a Bayesian point of view, by many authors; see, for example, Arnold et al. [6] and El-Din et al. [7]. Howlader and Hossain [8] presented Bayesian estimation of the survival function of the Lomax distribution. Ghitany et al. [9] considered Marshall-Olkin approach and extended Lomax distribution. Cramer and Schmiedt [10] considered progressively type-II censored competing risks data from Lomax distribution. The Lomax distribution has applications in economics, actuarial modelling, queuing problems and biological sciences; for details, we refer to Johnson et al. [11]. A positive random variable is said to have the Lomax distribution, abbreviated as , if it has the probability density function (pdf) Here, and are the shape and the scale parameters, respectively. The survival function (sf) associated with (1) is Further probabilistic properties of this distribution are given, for example, in Arnold [12]. This paper is concerned with the problem of estimating for Lomax based on general progressive censored data. The reliability of a component during a given period of time is defined as the probability that its strength exceeds the stress , and symbolically we write . We assume and to be independent, and each follows a Lomax distribution. A good overview on estimating can be found in the monograph of Kotz
References
[1]
A. Balkema and L. de Haan, “Residual life time at great age,” Annals of Probability, vol. 2, no. 5, pp. 792–804, 1974.
[2]
M. C. Bryson, “Heavy-tailed distributions: properties and tests,” Technometrics, vol. 16, no. 1, pp. 61–68, 1974.
[3]
M. Ahsanullah, “Record values of Lomax distribution,” Statistica Nederlandica, vol. 41, no. 1, pp. 21–29, 1991.
[4]
N. Balakrishnan and M. Ahsanullah, “Relations for single and product moments of record values from Lomax distribution,” Sankhya B, vol. 56, no. 2, pp. 140–146, 1994.
[5]
A. Childs, N. Balakrishnan, and M. Moshref, “Order statistics from non-identical right-truncated Lomax random variables with applications,” Statistical Papers, vol. 42, no. 2, pp. 187–206, 2001.
[6]
B. C. Arnold, E. Castillo, and J. M. Sarabia, “Bayesian analysis for classical distributions using conditionally specified priors,” Sankhya B, vol. 60, no. 2, pp. 228–245, 1998.
[7]
M. M. El-Din, H. M. Okasha, and B. Al-Zahrani, “Empirical bayes estimators of reliability performances using progressive type-II censoring from Lomax model,” Journal of Advanced Research in Applied Mathematics, vol. 5, no. 1, pp. 74–83, 2013.
[8]
H. A. Howlader and A. M. Hossain, “Bayesian survival estimation of Pareto distribution of the second kind based on failure-censored data,” Computational Statistics and Data Analysis, vol. 38, no. 3, pp. 301–314, 2002.
[9]
M. E. Ghitany, F. A. Al-Awadhi, and L. A. Alkhalfan, “Marshall-Olkin extended lomax distribution and its application to censored data,” Communications in Statistics, vol. 36, no. 10, pp. 1855–1866, 2007.
[10]
E. Cramer and A. B. Schmiedt, “Progressively type-II censored competing risks data from Lomax distributions,” Computational Statistics and Data Analysis, vol. 55, no. 3, pp. 1285–1303, 2011.
[11]
N. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distribution, vol. 1, John Wiley & Sons, New York, NY, USA, 2nd edition, 1994.
[12]
B. C. Arnold, Pareto Distributions, International Cooperative, Silver Spring, Md, USA, 1983.
[13]
S. Kotz, Y. Lumelskii, and M. Pensky, The Stress-Strength Model and Its Generalizations: Theory and Applications, World Scientific, New York, NY, USA, 2003.
[14]
A. Baklizi, “Likelihood and Bayesian estimation of Pr (X < Y) using lower record values from the generalized exponential distribution,” Computational Statistics and Data Analysis, vol. 52, no. 7, pp. 3468–3473, 2008.
[15]
M. Z. Raqab, M. T. Madi, and D. Kundu, “Estimation of P(Y < X) for the three-parameter generalized exponential distribution,” Communications in Statistics, vol. 37, no. 18, pp. 2854–2864, 2008.
[16]
D. Kundu and M. Z. Raqab, “Estimation of R = P(Y < X) for three-parameter Weibull distribution,” Statistics and Probability Letters, vol. 79, no. 17, pp. 1839–1846, 2009.
[17]
H. Panahi and S. Asadi, “Inference of stress-strength model for a lomax distribution,” Proceedings of World Academy of Science, Engineering and Technology, vol. 7, no. 55, pp. 275–278, 2011.
[18]
A. A. Soliman, A. Y. Al-Hossain, and M. M. Al-Harbi, “Predicting observables from Weibull model based on general progressive censored data with asymmetric loss,” Statistical Methodology, vol. 8, no. 5, pp. 451–461, 2011.
[19]
N. Balakrishnan and R. A. Sandhu, “A simple simulational algorithm for generating progressive type-II censored samples,” The American Statistician, vol. 49, no. 2, pp. 229–230, 1995.
[20]
B. Saraoglu, I. Kinaci, and D. Kundu, “On estimation of R = P(Y < X) for exponential distribution under progressive type-II censoring,” Journal of Statistical Computation and Simulation, vol. 82, no. 5, pp. 729–744, 2012.
[21]
N. Balakrishnan and R. Aggarwala, Progressive Censoring: Theory, Methods and Applications, Birkhauser, Boston, Mass, USA, 2000.
[22]
B. Nandi and A. B. Aich, “A note on confidence bounds for P(Y > X) in bivariate normal samples,” Sankhya B, vol. 56, no. 2, pp. 129–136, 1994.
[23]
A. Barbiero, “Interval estimators for reliability: the bivariate normal case,” Journal of Applied Statistics, vol. 39, no. 3, pp. 501–512, 2012.
[24]
F. J. Rubio and M. F. Steel, “Bayesian inference for P(X < Y) using asymmetric dependent distributions,” Bayesian Analysis, vol. 7, no. 3, pp. 771–792, 2012.
[25]
A. Zellner, “Bayesian and non-Bayesian estimation using balanced loss functions,” in Statistical Decision Theory and Methods, J. O. Berger and S. S. Gupta, Eds., pp. 337–390, Springer, New York, NY, USA, 1994.
[26]
M. J. Jozani, E. Marchand, and A. Parsian, “On estimation with weighted balanced-type loss function,” Statistics and Probability Letters, vol. 76, no. 8, pp. 773–780, 2006.
[27]
J. Ahmadi, M. J. Jozani, E. Marchand, and A. Parsian, “Bayes estimation based on k-record data from a general class of distributions under balanced type loss functions,” Journal of Statistical Planning and Inference, vol. 139, no. 3, pp. 1180–1189, 2009.
[28]
C. P. Robert and G. Casella, Monte Carlo Statistical Methods, Springer, New York, NY, USA, 2005.
[29]
S. K. Upadhyay and A. Gupta, “A bayes analysis of modified Weibull distribution via Markov chain Monte Carlo simulation,” Journal of Statistical Computation and Simulation, vol. 80, no. 3, pp. 241–254, 2010.
[30]
Z. F. Jaheen and M. M. Al Harbi, “Bayesian estimation for the exponentiated Weibull model via Markov chain Monte Carlo simulation,” Communications in Statistics: Simulation and Computation, vol. 40, no. 4, pp. 532–543, 2011.
