| In recent decades, the study on the empirical Bayes (EB) methods have been being paied closely attention to, because of the methods having extensive application background and prospect and having much more theoretical significance.Linear exponential distribution can be regarded as a gerenral exponential distribution, its hazard function is a linear function about time or age in linear exponential models, and it is one of the reasonable models for lifetime distributions of random phenomena. Thus, it is necessary to study and analysis its statistical property. From chapter 2 to chapter 7, empirical Bayes estimation and tests for parameter of linear exponential distribution are studied in this thesis.Suppose that the parameterθis given, random variable X has the linear exponential distribution density function as follows whereθis the parameter,assumeμ> 0is constant in this thesis. The sample space isΩ= { x x > 0},the parameter space is .In the case of identically independently distribution (i.i.d.) samples, empirical Bayes one-sided and two-sided tests for parameter of linear exponential distribution are discussed in chapter 2 and chapter 3, respectively, the Bayes rules and the empirical Bayes rules for parameter of linear exponential distribution are constructed by using the kernel-type density estimation. The asymptotically optimal ( a.o. ) property and convergence rates for the proposed empirical Bayes test rules are obtained under suitable conditions and convergence rates can arbitrary close to O ( n-1/2).At present, there are many papers about empirical Bayes tests problems in literature. It is easy to see that almost all papers studied EB tests in the case of i.i.d., the random samples are not always i.i.d. but correlate in practical problems such as reliability theory and penetration theory and some multivariable analysis, the negatively associated(NA) and the positively associated ( PA)samples are accepted. Thus, it is necessary to study EB tests in the correlate samples. Random variables ( r .v . ) X 1 , X 2, , X n( n≥2)are said to be negatively associated samples if for every pair of disjoint subsets A1 and A2 of {1,2, , n} , where f1 and f 2 are non-decreasing for each variable (or non-increasing for every variable ) such that this covariance exists. The r .v . series { X j, j∈N}is said to be negatively associated samples if for every nature number n≥2, X 1 , X 2, , X nare negatively associated.Empirical Bayes one-sided and two-sided tests for parameter of linear exponential distribution are studied in chapter 5 and chapter 6 in the case of NA samples, respectively, the Bayes rules and the empirical Bayes rules for parameter of linear exponential distribution are constructed by using the kernel-type density estimation. The asymptotically optimal property and convergence rates for the proposed empirical Bayes test rules are obtained under suitable conditions and convergence rates can arbitrary close to O ( n-1/2). At the end of chapter 5 and chapter 6, two examples are given to show that the suitable linear exponential distribution and prior distribution that satisfy the condition of theorems of chapter 5 and 6, respectively.In present, the discussions of EB estimation are almost all about to exponential distribution family and truncated distribution family, but linear exponential distribution is not exponential family and truncated family, thus, Empirical Bayes estimation for parameter of linear exponential distribution are studied in chapter 4 and chapter 7 that is new aspect. In chapter 4, suppose that the samples are i.i.d., the Bayes estimators and empirical Bayes estimators for the parameter of linear exponential distribution are constructed under the square loss function. It is shown that convergence rates of the proposed EB estimators can arbitrarily close O ( n-1/2) under suitable conditions.There are few literatures to discuss EB estimation in the case of NA samples. In chapter 7, firstly, the Bayes estimators for the parameter of linear exponential distribution are obtained under the square loss function,; secondly, the empirical Bayes estimators are constructed for the parameter of linear exponential distribution. It is shown that convergence rates of the proposed EB estimators can arbitrarily close O ( n-1 ) under suitable conditions.Pareto distribution is more and more paid attention to in many areas, essentially in economical areas. It is always used to describe such as personal incomes models, the living time of illness models, etc. Other models such as population of urban, development of natural phenomenon, waving of stock price, insure risk, etc; can be described by using Pareto distribution. Thus, it has important theoretical significance to study its statistical property. In chapter8, empirical Bayes one-sided and two-sided tests for parameter of Pareto distribution are discussed, respectively; the empirical Bayes rules for parameter of Pareto distribution are constructed by using the kernel-type density estimation in the case of NA samples. The asymptotically optimal property and convergence rates for the proposed empirical Bayes test rules are obtained under conditions of theorems.There are all kinds of trials data about life, survival time or failure time in survival analysis and reliability problems which are said to life data. One of trials is said to censored life tests, this kind test only carry to stop when the life of part samples terminal in all samples which been testing. Censor life tests conclude three kinds: type I censoring and type II censoring and random censoring. Type II censored tests stop trials when the numbers of lifetime samples reach to counts which are given at above. It has important significant to statistical analysis this kind of trials, because of it having extensive application background and prospect in survival analysis and reliability theory. In chapter 9, empirical Bayes two-sided tests problem for failure function of exponential distribution are studied. The empirical Bayes test rules are constructed by using the kernel-type density estimation, and convergence rates for the proposed empirical Bayes test rules are obtained under suitable conditions.Almost all studies about EB estimation are for the parameter of distribution but few for the parameter function of distribution. In chapter 10, the empirical Bayes estimator is derived under square loss function and empirical Bayes estimator for the hazard function of exponential distribution in the case of typeⅡcensored samples are constructed. It is shown that the proposed EB estimators are asymptotically optimal with convergence rates , where is a given integer.
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