Posterior Risks Research Articles

Mixture of transmuted Pareto distribution: Properties, applications and estimation under Bayesian framework

Transmuted distributions are flexible skewed families constructed by the induction of one or more additional parameters to a parent distribution. This paper investigates the potential usefulness of a two-component mixture of Transmuted Pareto Distribution (TPaD) under a Bayesian framework assuming type-I right censored sampling. For Bayesian analysis, noninformative as well as informative priors are assumed while three loss functions, namely the squared error loss function (SELF), precautionary loss function (PLF), and quadratic loss function (QLF) are considered to estimate the unknown parameters. Furthermore, Bayesian credible intervals (BCIs) are also discussed in this study. Since the posterior distributions do not have explicit forms, posterior summaries are computed using a Markov Chain Monte Carlo (MCMC) technique. The performance of the Bayes estimators is assessed by their posterior risks assuming different sample sizes and censoring rates. To highlight the practical significance of a two-component mixture of transmuted Pareto distribution (TPaD), a medical data set collected for rental calculi problem is analyzed in this study. Furthermore, annual flood rate data collected at the Floyd River are also discussed in this study.

Bayesian analysis for 3-component mixture of exponentiated Weibull distribution assuming non-informative priors

ABSTRACTBayesian analysis of 3-component mixture model, of exponentiated Weibull distribution under type-I right censoring scheme, is explored in this study. With the help of non-informative (uniform and Jeffreys) priors and loss functions (e.g. squared error loss function, quadratic loss function, precautionary loss function and DeGroot loss function), Bayes estimators and posterior risks are derived. The Bayes estimators and posterior risks are observed as a function of the test termination time. Simulation study as well as a practical example is given in this study.

ON THE BAYESIAN ANALYSIS OF EXTENDED WEIBULL-GEOMETRIC DISTRIBUTION

The paper deals with the Bayes estimation of Extended Weibull-Geometric (EWG) distribution. In particular, we discuss Bayes estimators and their posterior risks using the noninformative and informative priors under different loss functions. Since the posterior summaries cannot be obtained analytically, we adopt Markov Chain Monte Carlo (MCMC) technique to assess the performance of Bayes estimates for different sample sizes. A real life example is also part of this study.

On the Bayesian Analysis of Two-Component Mixture of Transmuted Weibull Distribution

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.

Early prediction of preeclampsia and small-for-gestational-age via multi-marker model in Chinese pregnancies: a prospective screening study

BackgroundRecent evidence suggests early screening of preeclampsia and small-for-gestational-age (SGA) would benefit pregnancies followed by subsequent prophylactic use of aspirin. Multi-marker models have shown capability of predicting preeclampsia and SGA in first trimester. Yet the clinical feasibility of combined screening model for Chinese pregnancies has not been fully assessed. The aim of this study is to evaluate the applicability of a multi-marker screening model to the prediction of preeclampsia and SGA in first trimester particularly among Chinese population.MethodsThree thousand two hundred seventy pregnancies meeting the inclusion criteria took first-trimester screening of preeclampsia and SGA. A prior risk based on maternal characteristics was evaluated, and a posterior risk was assessed by combining prior risk with multiple of median (MoM) values of mean arterial pressure (MAP), serum placental growth factor (PLGF) and pregnancy associated plasma protein A (PAPP-A). Both risks were calculated by Preeclampsia PREDICTOR™ software, Perkin Elmer. Screening performance of prior and posterior risks for early and late preeclampsia by using PREDICTOR software was shown by Receiver Operating Characteristics (ROC) curves. The estimation of detection rates and false positive rates of delivery with both preeclampsia and SGA was made.ResultsEight cases developed early preeclampsia (0.24%) and 35 were diagnosed as late preeclampsia (1.07%). Five with early preeclampsia and ten with late preeclampsia later delivered SGA newborns (0.46%); 84 without preeclampsia gave birth to the SGAs (2.57%). According to ROC curves, posterior risks performed better than prior risks in terms of preeclampsia, especially in early preeclampsia. At 10% false positive rate, detection rates of early and late preeclampsia were 87.50 and 48.57%, detection rates of early and late SGA were 41.67 and 28.00%, respectively. For SGA, detection rates in cases with preeclampsia were much higher than those in absence of it.ConclusionsThis study demonstrates that combined screening model could be useful for predicting early preeclampsia in Chinese pregnancies. Furthermore, the performance of SGA screening by same protocol is strongly associated with preeclampsia.

Bayesian estimation and prediction for Burr‐Rayleigh mixture model using censored data

SummaryIn this study, Burr‐XII and Rayleigh distributions are combined to form a new mixture model that is considered to model heterogeneous data. Our objective is to estimate parameters of the proposed mixture model using Bayesian technique under type‐I censoring. Bayesian parameter estimation for the said mixture model is conducted by using informative priors, ie, gamma and squared root inverted gamma (SRIG) as well as noninformative prior, ie, Jeffrey's prior. Squared error loss function (SELF) and quadratic loss function (QLF) are employed to obtain and Bayes estimators. Properties of the proposed Bayes estimators are highlighted through a simulation study. When prior distributions and loss functions utilized in the study are compared in terms of posterior risks, informative prior found to be more suitable and decision turns out to be in favor of QLF. Prediction limits for the single sample case and two sample case are obtained to provide an insight into future sample data. Application of the proposed model is also elaborated using a real‐life example.

Bayesian estimation of the mixture of Burr Type-XII distributions using doubly censored data

This study discusses the Bayesian and maximum likelihood estimation methods for analyzing the data from a 3-component mixture of Burr Type-XII probability distributions. The maximum likelihood estimators with their variances cannot be obtained in an explicit form and thus an iterative procedure is used to calculate them numerically. Contrary to this, elegant closed form algebraic expressions of Bayes estimators and their posterior risks are derived. Using the informative and noninformative priors, the posterior predictive distributions along with predictive intervals are also discussed. A method of eliciting hyperparameters using prior predictive distribution is also a part of this study. Some interesting properties (including, posterior risks and Bayesian predictive intervals) of Bayes estimators and their behavior across different sample sizes, left and right test termination times, informative prior versus Jeffreys noninformative prior, are provided via a detailed Monte Carlo simulation study. To assess the suitability and application of the proposed model, a real life data example is also discussed in this article. Based on the simulated results and real data application, it is concluded that the IP paired with DLF (SELF) is a more suitable for estimating mixing component.

Parameter and reliability estimation of inverted Maxwell mixture model

In present study the probability density function of mixture is derived for inverse Maxwell density. The main distributional properties and reliability characteristics are studied. Maximum likelihood estimation of the pertinent parameters along with failure rate functions and reliability are obtained. The Bayesian study of anonymous parameters of inverse Maxwell mixture model, assuming three priors is considered employed distinct loss functions. The prior reliance of mixture density is characterized by the inverted gamma, uniform and Jeffreys prior. The efficiencies of the considered set of estimates of mixture distribution parameters are studied through simulation. To scrutinize the response of prior reliance and loss functions posterior risks of the Bayes estimators are figured out and differentiated. Bayes estimator assuming the informative have been observed performing better.

Bayesian inference from the mixture of half-normal distributions under censoring

This study considers the Bayesian inference for the mixture of two components of half-normal distribution using non-informative and informative prior. Several of its structural properties were derived, including explicit expression for mean, median, mode, reliability and hazard rate functions. Due to cost and time constraints, in most lifetime testing experiments censoring is an obligatory feature of lifetime datasets. We investigated Bayesian estimation of the parameters using various loss functions. The prior belief of the mixture model is represented by the uniform and square-root inverted gamma priors. Some properties of the model with graphs of the mixture density and hazard function are also discussed. The efficiencies of the proposed set of estimates of the mixture model parameters were studied through simulation and a real life dataset. Posterior risks of the Bayes estimators are evaluated and compared to explore the effect of prior belief and loss functions.

A Family of Bayes Estimators for the Parameters of the Generalized Gamma Distribution

The paper aims to propose a family of estimators for the Bayesian analysis of three parametric generalized gamma (GG) distribution under different priors and loss functions. We have proposed the Gibbs sampler to obtain the numerical solutions for the point and interval estimators of the parameters using WinBugs. The comparison among the different estimators has been made in terms of posterior risks and the widths of the corresponding credible intervals. A simulation study has been conducted to investigate the performance of the estimators under different combinations of the parametric values and using various sample sizes. A real life data set has been analyzed to illustrate the practical applicability of the results.

Bayesian Analysis of a Shape Parameter of the Weibull-Frechet Distribution


 
 
 
 In this paper, we estimate a shape parameter of the Weibull-Frechet distribution by considering the Bayesian approach under two non-informative priors using three different loss functions. We derive the corresponding posterior distributions for the shape parameter of the Weibull-Frechet distribution assuming that the other three parameters are known. The Bayes estimators and associated posterior risks have also been derived using the three different loss functions. The performance of the Bayes estimators are evaluated and compared using a comprehensive simulation study and a real life application to find out the combination of a loss function and a prior having the minimum Bayes risk and hence producing the best results. In conclusion, this study reveals that in order to estimate the parameter in question, we should use quadratic loss function under either of the two non-informative priors used in this study.
 
 
 
 

Characterization and estimation of the length biased Nakagami distribution

In this paper, we introduce the length biased form of the Nakagami distribution known as length biased Nakagami distribution (LBND). Some properties of this model were studied such as the moments, reliability function, and the hazard rate function. Maximum likelihood and Bayes estimates of the scale parameter are derived. Also the Posterior risks under different loss functions are obtained. The results are applied on the data regarding the life test of 59 conductors.

Approximate Bayesian analysis of doubly censored samples from mixture of two Weibull distributions

The purpose of the paper is to estimate the parameters of the two-component mixture of Weibull distribution under doubly censored samples using Bayesian approach. The choice of Weibull distribution is made due to its (i) capability to model failure time data from engineering, medical and biological sciences (ii) added advantages over the well-known lifetime distributions such as exponential, Raleigh, lognormal and gamma distribution in terms of flexibility, increasing and decreasing hazard rate and closed-form distribution function and hazard rate. The proposed two-component mixture of Weibull distribution is even more flexible than its conventional form. However, the estimation of the parameters from the proposed mixture is more complex. Further, we have assumed couple of loss functions under non informative prior for the Bayesian analysis of the parameters from the mixture model. As the resultant Bayes estimators and associated posterior risks cannot be derived in the closed form, we have used the importance sampling and Lindley’s approximation to obtain the approximate estimates for the parameters of the mixture model. The comparison between the performances of approximation techniques has been made on the basis of simulation study and real-life data analysis. The importance sampling is found to be better than Lindley’s approximation as it gives better estimation for shape and mixing parameters of the mixture model and computations under this technique are much easier/shorter than those under Lindley’s approximation.

Hierarchical Bayesian modeling of spatio-temporal patterns of scrub typhus incidence for 2009–2013 in South Korea

This study aims to analyze spatio-temporal patterns of scrub typhus incidence and to identify environmental risk factors in South Korea from 2009 to 2013 using hierarchical Bayesian Poisson model. We classified 244 municipal areas into nine categories based on the posterior relative risks and their temporal trends and these patterns were visualized. Environmental covariates (i.e. land surface temperature, precipitation and vegetation) were compiled by season to reflect the seasonal behavior of trombiculid mites and rodents and by year to explain year-based environmental changes. A total of six covariates were incorporated in the regression model and environmental effects were evaluated. We found areas in the south, southwest, and southeast of South Korea that present relatively high risk, and the north and southeast parts show rapid increases during the study period. The spatio-temporal maps present similar patterns as well. A positive correlation is found between the scrub typhus risk and precipitation in summer, and normalized difference vegetation index in fall. In contrast, the scrub typhus risk is negatively correlated with land surface temperature in fall. This information has implications for prioritizing disease control resources on high-risk areas. Furthermore, the regression results might be utilized for predicting scrub typhus incidence and contribute to the design of disease control programs in the future.

Three-Component Mixture of Rayleigh Model Under Doubly Censored Samples: A Bayesian Look

Feroze and Aslam (J Natl Sci Found Sri Lanka 42:325–334, 2014) and Sindhu et al. (J Mod Appl Stat Methods 13:259–286, 2014) have considered the Bayesian analysis of the two-component mixture of lifetime distributions under doubly censored samples. In this paper, the concept has been extended for the three-component mixture of lifetime (Rayleigh) distribution under doubly censored samples. The likelihood function for the mixture of lifetime models under doubly type-II censored samples has been introduced. Different priors and loss functions have been assumed for the derivation of Bayes estimators and posterior risks. The comparisons among various estimators have been made by analyzing the simulated and real data sets.

Bayesian Estimation for 3-Component Mixture of Generalized Exponential Distribution

This paper presents the Bayesian analysis of 3-component mixture of generalized exponential distribution. The Bayesian analysis and maximum likelihood estimation of five parameters have been performed by assuming type-I right censored data. Monte Carlo simulation has been adopted for the comparison of Bayes estimates, posterior risks, maximum likelihood estimates and maximum likelihood risks. Furthermore, the study assesses the performance of Bayes and maximum likelihood estimates by using different sample sizes, proportion of mixture components, censoring rates and loss functions. The Bayes estimates are examined by using non-informative Jeffreys and uniform prior under square error loss function, precautionary loss function and DeGroot loss function. The maximum likelihood risks are obtained by using the Fisher information matrix.

Statistical properties and different methods of estimation of Gompertz distribution with application

This article addresses the various properties and different methods of estimation of the unknown parameters of Gompertz distribution. Although, our main focus is on estimation from both frequentist and Bayesian point of view, yet, various mathematical and statistical properties of the Gompertz distribution (such as quantiles, moments, moment generating function, hazard rate, mean residual lifetime, mean past lifetime, stochasic ordering, stress-strength parameter, various entropies, Bonferroni and Lorenz curves and order statistics) are derived. We briefly describe different frequentist approaches, namely, maximum likelihood estimators, moments estimators, pseudo-moments estimators, modified moments estimators, L-moment estimators, percentile based estimators, least squares and weighted least squares estimators, maximum product of spacings estimators, minimum spacing absolute distance estimators, minimum spacing absolute-log distance estimator, Cramér-von-Mises estimators, Anderson-Darling and right-tail Anderson-Darling and compare them using extensive numerical simulations. Coverage probabilities for the frequentist methods are also obtained. Next we consider Bayes estimation under different types of loss function (symmetric and asymmetric loss functions) using gamma priors for both shape and scale parameters. Furthermore, the Bayes estimators and their respective posterior risks are computed and compared using MCMC algorithm. Finally, a real data set have been analyzed for illustrative purposes.

Bayesian Analysis of Heterogeneous Doubly Censored Lifetime Data using the 3-Component Mixture of Rayleigh Distributions: A Monte Carlo Simulation Study

This article is about Bayesian estimation of parameters of a heterogeneous 3-component mixture of Rayleigh distributions (3-CMRD) generating a mixture data. Being the most popular and reasonable sampling scheme in reliability and survival analyses, the doubly censored sampling scheme is considered. The Bayes estimators and their posterior risks are derived under various situations. In addition, elicitation of hyperparameters is presented. Algebraic expressions for posterior predictive distribution and Bayesian predictive intervals are derived. Assuming the informative and the non-informative priors, a comprehensive Monte Carlo simulation is conducted to examine the performance of the Bayes estimators under symmetric and asymmetric loss functions. Finally, to highlight the practical importance, the proposed 3-compnent mixture model is applied to a doubly censored lifetime data from a real life situation. It is observed that the analysis of doubly censored data in Bayesian framework, the SRIGP paired with SELF (DLF) is suitable choice for estimating mixing proportion (component) parameters.

Bayesian Estimation of 3-Component Mixture of the Inverse Weibull Distributions

This article focuses on the study of a 3-component mixture of the inverse Weibull distributions under Bayesian perspective. The censored sampling scheme is used because it is popular in reliability theory and survival analysis. To achieve this objective, the Bayes estimates of the parameter of the mixture model along with their posterior risks using informative and non-informative priors are attained. These estimates have been acquired under two cases: (a) when the shape parameter is known and (b) when all parameters are unknown. For the case (a), Bayes estimates are gained under three loss functions while for the case (b) only the squared error loss function is used. To study numerically, the performance of the Bayes estimators under different loss functions, their statistical properties have been simulated for different sample sizes and test termination times.

On estimation of Frechet distribution with known shape using Bayesian analysis under informative priors

In this study, Frechet distribution has been studied by using Bayesian analysis. Posterior distribution has been derived by using gamma and exponential. Bayes estimators and their posterior risks has been derived using five different loss functions. Elicitation of hyperparameters has been done by using prior predictive distributions. Simulation study is carried out to study the behavior of posterior distribution. Quasi quadratic loss function and exponential prior are found better among all.