Abstract
Bayesian inference of the 3-component unit Lindley right censored mixture is presented in this paper. The posterior distributions of the parameters are derived assuming informative (gamma) as well as noninformative (uniform and Jeffreys) priors. For the gamma (informative) prior, hyperparameters are elicited using prior predictive distribution. The Bayesian estimation has been carried out considering both symmetric and asymmetric loss functions (squared error, quadratic, weighted, and precautionary loss functions). Simulation studies for various sample sizes and different threshold values (test termination times) are considered to evaluate the performances of the Bayes estimators w.r.t their posterior risks under the said loss functions. Real life flood data from Naser Lake is also analyzed as a 3-component mixture for the sake of illustrative purpose. The simulation study and data analysis reveals that the estimates under informative prior perform better than the noninformative priors. Also, it is observed that the increase in sample size and the threshold value (test termination time) are inversely proportional to the posterior risks. Among the loss functions considered, the loss functions performance from the best to the least, w.r.t the posterior risk, is as follows: precautionary loss function < weighted loss function < quadratic loss function < squared error loss function. Posterior risks are directly proportional to the sample size and threshold value (test termination time).
Highlights
Extreme value models are typically used to describe the distribution of rare events and a plethora of recent articles have proposed various extreme value mixture models for threshold estimation and quantifying the corresponding uncertainty. e convenience of employing mixture models to extreme value data essentially arises from the ease with which multiple datasets and forecasting situations can be premeditated using such models
Mixture models facilitate the use of versatile modeling in real life areas, where only simple models were considered until the end of the twentieth century. e flexibility of mixture models has proliferated their exploitation as a convenient and semiparametric tool to model unknown distributional shapes
Before the dawn of online resources, only two types of books used to be available to readers: the paperback and hardback books. e paperback are comparatively cheaper than hardbacks. erefore, a 2component mixture distribution can sufficiently model the prices
Summary
Despite the advent of computational and numerical techniques, complexity of beta model has always been an issue that hindered the extensive analysis of variables bounded over the [0, 1] region. e quest to find a better and comparatively less cumbersome probability model led the researchers towards Kumaraswamy distribution introduced by Kumaraswamy [1]. e pursuit of a better and less complicated model led to the single parameter unit Lindley distribution proposed by Mazucheli et al [2] as an alternate to analyze the life time data dealing with proportions, indices, fractions, or the characteristics bounded over the unit interval. e analysis of statistical properties demonstrated that it yielded better fit than the most widely used beta distribution. ey further established that unit Lindley distribution is the only single parameter distribution, amongst all the distributions bounded over interval (0, 1), which is unimodal and has closed form expressions for c.d.f, moments, and quantiles. E pursuit of a better and less complicated model led to the single parameter unit Lindley distribution proposed by Mazucheli et al [2] as an alternate to analyze the life time data dealing with proportions, indices, fractions, or the characteristics bounded over the unit interval. Ey further established that unit Lindley distribution is the only single parameter distribution, amongst all the distributions bounded over interval (0, 1), which is unimodal and has closed form expressions for c.d.f, moments, and quantiles. Akdur [6] discussed parameter estimation unit Lindley mixed effect model in case of clustered and longitudinal proportion data. Most of the previous analysis of finite mixture models, Bayesian or otherwise, considered 2-component mixtures. Motivated by the previous research on 3-component mixture distributions, we examine the 3-component unit Lindley mixture under Bayesian inference.
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