Comparison of the Bayesian and maximum likelilhood estimation for Weibull distribution

Problem statement: The Weibull distribution has been widely used especially in the modeling of lifetime event data. It provides a statistical model which has a wide variety of applications in many areas, and the main advantage is its ability in the context of lifetime event, to provide reasonably accurate failure analysis and failure for ecasts especially with extremely small samples. The conventional maximum likelihood method is the usual way to estimate the parameters of a distribution. Bayesian approach has received much attention and in contention with other estimation methods. In this study we explore and compare the performance of the maximum likelihood estimate with the Bayesian estimate for the Weibull distribution. Approach: The maximum likelihood estimation, Bayesian using Jeffrey prior and the extension of Jeffrey prior information for estimating the parameters of Weibull distribution of life time are presented. We explore the performance of these estimators numerically under varyin g conditions. Through the simulation study comparison are made on the performance of these estimators with respect to the Mean Square Error (MSE) and Mean Percentage Error(MPE). Results: For all the varying sample size, several specific values of the scale parameter of the Weibull distribution and for the values specify for the extension of Jeffrey prior, the estimators of the maximum likelihood method result in smaller MSE and MPE compared to Bayesian in majority of the cases. Nevertheless in all cases for both methods the MSE and MPE decrease as sample size increases. Conclusion: Based on the results of this simulation study the Bayesian approach used in the estimating of Weibull parameters is found to be not superior compared to the conventional maximum likelihood method with respect to MSE and MPE values

Estimation of the Weibull Distribution Parameters and Reliability Using Kernel and Bayes Approaches

A new estimation technique based on the non-parametric kernel density estimation is introduced as an alternative and reliable technique for estimation in life testing models. This technique estimates the density functions of the parameters and reliability directly from the data without any prior assumptions about the underlying distribution parameters. The efficiency of this technique has been studied comparing to the Bayesian estimation of the parameters and reliability of the Weibull distribution based on the non-informative, informative and the informative conjugate priors, via Monte Carlo simulations, which indicated the robustness of the proposed method than the Bayesian approach. Finally, a numerical example is given to illustrate the densities and the inferential methods developed in this paper

Bayesian Life Test Planning for the Log-Location-Scale Family of Distributions

This paper describes Bayesian methods for life test planning with censored data from a log-location-scale distribution, when prior information of the distribution parameters is available. We use a Bayesian criterion based on the estimation precision of a distribution quantile. A large sample normal approximation gives a simplified, easy-tointerpret, yet valid approach to this planning problem, where in general no closed form solutions are available. To illustrate this approach, we present numerical investigations using the Weibull distribution with Type II censoring. We also assess the effects of prior distribution choice. A simulation approach of the same Bayesian problem is also presented as a tool for visualization and validation. The validation results generally are consistent with those from the large sample approximation approach

Estimation of Inverse Weibull Distribution Under Type-I Hybrid Censoring

The hybrid censoring is a mixture of Type I and Type II censoring schemes. This paper presents the statistical inferences of the Inverse Weibull distribution when the data are Type-I hybrid censored. First we consider the maximum likelihood estimators of the unknown parameters. It is observed that the maximum likelihood estimators can not be obtained in closed form. We further obtain the Bayes estimators and the corresponding highest posterior density credible intervals of the unknown parameters under the assumption of independent gamma priors using the importance sampling procedure. We also compute the approximate Bayes estimators using Lindley's approximation technique. We have performed a simulation study and a real data analysis in order to compare the proposed Bayes estimators with the maximum likelihood estimators.Comment: This paper is under review in the Austrian Journal of Statistics and will likely be published ther

Hierarchical models for semi-competing risks data with application to quality of end-of-life care for pancreatic cancer

Readmission following discharge from an initial hospitalization is a key marker of quality of health care in the United States. For the most part, readmission has been used to study quality of care for patients with acute health conditions, such as pneumonia and heart failure, with analyses typically based on a logistic-Normal generalized linear mixed model. Applying this model to the study readmission among patients with increasingly prevalent advanced health conditions such as pancreatic cancer is problematic, however, because it ignores death as a competing risk. A more appropriate analysis is to imbed such studies within the semi-competing risks framework. To our knowledge, however, no comprehensive statistical methods have been developed for cluster-correlated semi-competing risks data. In this paper we propose a novel hierarchical modeling framework for the analysis of cluster-correlated semi-competing risks data. The framework permits parametric or non-parametric specifications for a range of model components, including baseline hazard functions and distributions for key random effects, giving analysts substantial flexibility as they consider their own analyses. Estimation and inference is performed within the Bayesian paradigm since it facilitates the straightforward characterization of (posterior) uncertainty for all model parameters including hospital-specific random effects. The proposed framework is used to study the risk of readmission among 5,298 Medicare beneficiaries diagnosed with pancreatic cancer at 112 hospitals in the six New England states between 2000-2009, specifically to investigate the role of patient-level risk factors and to characterize variation in risk across hospitals that is not explained by differences in patient case-mix