We develop a new empirical Bayes analysis in multiple regression models. In the present work we consider multivariate skew normal as prior for coefficients of the model in a skew-normal population and give empirical Bayes Estimation for parameters of the model. The marginal distribution of response is found to be a closed skew-normal distribution. The empirical Bayes estimator is found in a closed form and the model is applied on a data set.

In risk analysis based on Bayesian framework, premium calculation requires specification of a prior distribution for the risk parameter in the heterogeneous portfo-lio. When the prior knowledge is vague, the E-Bayesian and robust Bayesian analysis can be used to handle the uncertainty in specifying the prior distribution by consid-ering a class of priors instead of a single prior. In this paper, we study the E-Bayes and robust Bayes premium Estimation and prediction in exponential model under the squared log error loss function. A prequential analysis in a simulation study is carried out to compare the proposed predictors. Finally, a real data example is included for illustrating the results.

This paper deals with the analysis of log-Linear models for multi-way contingency tables utilizing the empirical Bayes procedure. We consider such a table as a multi-nomial distribution involving a vector of parameters.A log-Linear representation is introduced. Assuming a Dirichlet prior for the parameters of the log-Linear model, we employ the Bayesian paradigm to infer the structure of the table. Assuming the exact prior distribution is not known, the empirical Bayes procedure is utilized to estimate the prior. Based on the estimated posterior, the Bayes estimator and its distributional properties are studied. Asymptotic normality of the estimators is established. The procedure is illustrated by an example.

In this paper, we introduced straightforward formulas for the Bayes risk linked to the Linex loss function, which we then applied to estimate parameters of the normal, Poisson, and fractional Weibull distributions. We aimed to investigate the development of a Linear Bayes estimator using the Linex loss function and successfully derived it for the normal and Poisson scenarios. We also demonstrated the process of creating empirical Bayes estimates using Linex loss and applied it to observed frequencies fn(x) produced by the Poisson-gamma model.

Let S be the matrix of residual sum of square in Linear modelY=Ab+e, where the matrix of errors is distributed as elliptically contoured with unknown scale matrix S. For Stein loss function, L1 (S ,S) =tr (SˆS-1) -log |(SˆS-1|-p, and squared loss function, L2 ((SˆS-1) =tr ((SˆS-1-I) 2, we offer empirical Bayes estimators of S, which dominate any scalar multiple of S, i.e., aS, by an effective amount. In fact, this study somehow shows that improvement of the empirical Bayes estimators obtained under the normality assumption remains robust under elliptically contoured model.

In this study, the application of Bayes networks and fault tree analysis in reliability Estimation have been investigated. Fault tree analysis is one of the most widely used methods for estimating reliability. In recent years, a method called "Bayes Network" has been used, which is a dynamic method, and information about the probable failure of the system components will be updated according to the components data. In other words, the Bayes network is generally distributed with primary values and relations between variables. In this research, the reliability of the automatic Environmental detection and fire extinguisher system by water sprinkler system has been estimated by FTA and Bayes NETWORKS, also their comparison has been investigated. The probabilistic calculation and the graphical drawing required for this research has been completed by MATLAB and MICROSOFT Bayes NETWORK software. Finally, the reliability of an automatic water sprinkler system was calculated 0. 93 by FTA. By updating the probability of failure and success, this value was changed to 0. 89. The reason for the difference can be interpreted by taking into account the third state of the system, i. e., functioning with lower efficiency. The lowest level of reliability relates to a diesel generator that acts as a standby member in the event of a power failure. At the end, suggestions for improving the reliability of the automatic fire sprinkler system provided.