Bayesian Estimation of Transmuted Pareto Distribution for Complete and Censored Data
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Bayesian Estimation of Transmuted Pareto Distribution for Complete and Censored Data Muhammad Aslam1 · Rahila Yousaf1 · Sajid Ali2 Received: 26 April 2020 / Revised: 21 July 2020 / Accepted: 22 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Transmuted distributions belong to the skewed family of distributions which are more flexible and versatile than the simple probability distributions. The focus of this article is the Bayesian estimation of three-parameter Transmuted Pareto distribution. In particular, we assumed noninformative and informative priors to obtain the posterior distributions. Bayesian point estimators and the associated precision measures are investigated under squared error loss function, precautionary loss function, and quadratic loss function. In addition to this, the Bayesian credible intervals are also computed under different priors. A simulation study using a Markov Chain Monte Carlo algorithm assuming uncensored and censored data in terms of different sample sizes and censoring rates is also a part of this study. The performance of Bayesian point estimators is assessed in term of posterior risks. Finally, two real life data sets of cardiovascular disease patients and of exceedances of Wheaton River flood are discussed in this article. Keywords Transmuted Pareto distribution · Loss functions · Bayes estimators · Posterior risks · Uniform prior · Informative prior · BCIs · MCMC · Censoring and Chi square test
* Sajid Ali [email protected] Muhammad Aslam [email protected] Rahila Yousaf [email protected] 1
Department of Mathematics and Statistics, Riphah International University, Islamabad, Pakistan
2
Department of Statistics, Quaid-i-Azam University, Islamabad, Pakistan
13
Vol.:(0123456789)
Annals of Data Science
1 Introduction Pareto distribution also known as the power law probability distribution is used to model different real life phenomena, including the distribution of income data, reliability, finance and actuarial sciences, economics, and sizes of firms [1–3]. Abdel-All et al. [4] presented the geometrical properties of the Pareto distribution while Ismail [5] discussed a simple estimator for the shape parameter of the Pareto distribution. Assuming the shape parameter α and the scale parameter β, the cumulative distribution function (CDF) of a Pareto random variable X is given by: ( )𝛼 𝛽 G(x) = 1 − (1) x whereas the probability density function (PDF) is:
g(x;𝛼, 𝛽) =
𝛼𝛽 𝛼 , x𝛼+1
x > 𝛽, 𝛼 > 0 , 𝛽 > 0
(2)
To generate skewed distributions, an approach based on the quadratic rank transmutation (QRM) was proposed by Shaw and Buckley [6] and since then, several families of distributions have been extended by inducting a parameter to the baseline model of continuous distributions [2, 7–11]. It is noticed from the literature that the transmuted families of distributions are more flexible in modeling and analyzing real life data in many diverse areas such as biostatistics, reliability, and survival analysis [12]. The PDF of a
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