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.

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.

The most widely used model in small area estimation is the area level or the Fay-Herriot model. In this model, it is typically assumed that both the area level random effects (model errors) and the sampling errors have a Gaussian distribution. However, considerable variations in error components (model errors and sampling errors) can cause poor performance in small area estimation. In this paper, to overcome this problem, the symmetric ,-stable distribution is used to deal with outliers in the error components. The model parameters are estimated with the empirical Bayes method. The performance of the proposed model is investigated in different simulation scenarios and compared with the existing classic and robust empirical Bayes methods. The proposed model can improve estimation results, in particular when both error components are normal or have heavy-tailed distribution.

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 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.

Introduction In classical methods of statistics, the parameter of interest is estimated based on a random sample using natural estimators such as maximum likelihood or unbiased estimators (sample information). In practice, the researcher has a prior information about the parameter in the form of a point guess value. Information in the guess value is called as nonsample information. Thompson (1968) proposed a shrinkage estimator by combining sample and nonsample information which is a linear combination of a natural estimator and the guess value. ...

Small area estimation has received a lot of attention in recent years due to necessity demand for reliable small area statistics. Direct estimator may not provide adequate precision, because sample size in small areas is seldom large enough. Hence, by employing models that can use auxiliary information and area effects in descriptions, one can increase the precision of direct estimators. Due to more readily available level auxiliary information of area, simplicity and possibility of evaluation of the assumptions used by survey data, area level model has become of comprehensive importance, nowadays. Therefore, basic area level models have been extensively studied in this paper to derive empirical best linear unbiased prediction (EBLUP), empirical Bayes (EB), and hierarchical Bayes (HB) with several different assumptions on parameters. Those models are used to option the small area estimators, i.e., the mean of household income in several provinces of Iran, including Khorasan-e- Razavi, Hamedan, Lorsetan, and Tehran. To assess small area estimators, we used 1700 urban households who live in those provinces from the data set of the 2006-2007 Household's Income and Expenditure Survey. Some sampling scheme has been utilized. The optimal total sample size has been more than 400 units, but we have only 212 units available. Due to shortage of sample size, we face large MSE's, encountered us with small area problem. There are three measures for comparison of small area methods, including average square error (ASE), average absolute of relative bias (AARB), and average of absolute bias (AAB).We have used two types of transformations, logarithm transformation, and Box-Cox transformation, because of abnormality and heterogeneity of variances.Our data analysis has shown that it is better to use Box-Cox transformation than to use logarithm transformation, i.e., the test statistic is more significant by using this transformation; but Box-Cox transformation causes large sampling variance, which in some cases results in non-convergence in Gibbs algorithm.Likewise, HB approach gives better results than EBLUP and EB. All of these approaches are better than direct estimator, i.e., they have smaller values of ASE, AASB, and AAB.

In this article, an attempt has been made to estimate the amount of sound transmission loss in a flat oval channel by applying the approach of statistical energy analysis. Correct estimation of sound transmission loss in an air conditioning channel is of great importance due to the harmful effects of noise pollution in the environment on human health. Simulation with the statistical energy analysis method is a powerful approach to estimate sound and vibration in problems in which we deal with complex and multi-part systems; is considered. In this method, first, a system is divided into several subsystems, and then by writing a matrix equation that includes the energy exchanges between subsystems and energy loss coefficients; It is investigated from the perspective of vibration and sound estimation.On average, the model presented in this research is able to estimate the sound transmission loss in different dimensions of the air conditioning channels according to the experimental results in the accuracy range of ± 2.5 dB. Considering that it seems that the results obtained from modeling with this method are in good agreement with the experimental data; The results of this research can be used as an efficient approach to estimate noise in oval shaped channels stretched in different lengths.