Dependency Case, The Empirical Bayes Statistical Analysis And Applied Research

In this paper, Bayesian theory and empirical Bayesian theory are used to study the parameter estimation of the distribution families. Using empirical Bayes methods , the parameters for positive exponential family and scale exponential family are discussed in the case of PA samples. Moreover, one-parameter distribution family parameter is discussed as the samples are m dependent. In the end, the reliability performances for series and parallel systems with n dependent components are studied.1. By using the kernel-type density estimation in the case of identically distributed and positively associated (PA) samples, the empirical Bayes (EB) test rules for the parameter of a positively exponential family are constructed and the asymptotic optimality is obtained. It is shown that the convergence rates of the proposed EB test rules can arbitrarily close to O(n^-(?)) under certain condition.2. For the scale-exponential family, the empirical Bayes estimation of the parameter are exhibited under asymmetric linex loss function. Combining PA samples, the EB estimators are discussed and the rates of convergence are also obtained.3. Based on m dependent samples, the linear EB estimation of the parameter is discussed . also the asymptotically optimal rates of convergence O(N^-(?)/(?)C_N^-(?)) is obtained.4. For the reliability parameters in n non-independent series and parallel systems with dependent time failure rates, the corresponding failure models are constructed. Meanwhile, we give the Bayesian estimation of reliability performances such as system parameters, reliability and failure rates.