Estimation Under Q-Symmetric Loss Function Of The Tail Probability And Scale Parameter Of The Exponential Distribution

The exponential distribution which is the earliest model in life research plays aimportant role in life test and the research of reliability. And its statistical methods havebeen widely developed. As to an exponential population with the distribution density:the parameter of e?λt0 (t0 > 0) is the tail probability. In practical application, it appearsas a reliability; therefore, there are a variety of its estimators. We make our researchin Chapter 1 on the estimation of e?λt0 (t0 > 0) under the q-symmetric entropy lossfunction:In the first section of the Chapter, we deduce our conclusion on the basis of a simpleexponential sample X = (X1,X2,···,Xn) as below:If the prior distribution of the scale parameterλisΓ(α,β) (α> 0,β> t0q),e?λt0 (t0 > 0) has an unique Bayes estimator in the sense of almost surely equality asfollows:whereWe generalize this estimator to the caseα≥0, -∞<β< +∞, and then takeinto account its equivalentwhere In the second section of the chapter, we continue our job on the admissibility ofδ(T) and obtain the conclusions below:Theorem 1.2.1 In the case that a > 2nq and b > 0, the estimator is admissible.Theorem 1.2.2 In the case that b > 0 and a = 2nq, the estimatorδ= is admissible.Theorem 1.2.3 In the case that b = 0 and a≥nq, the estimator is inadmissible.Theorem 1.2.4 In the case that b = 0 and 2nq < a < nq, the estimator is admissible.Theorem is admissible.Theorem the estimator inadmissible.In the third section of the chapter, we make our research on the Minimax estimatorsof e?λt0 under q-symmetric entropy function. We find that the Minimax estimator ofe?λt0 does not exist as the following theorem describes:Theorem 1.3.1 Under q-symmetric entropy function, none of the non-stochasticestimators of e?λt0 is the Minimax estimator.In the last section of the chapter, we study the properties of strong consistency andasymptotic normality of the Bayes estimators of e?λt0 (t0 > 0),and gain the conclusions below: Theorem As to any the estimatoris a strongly consistent estimator ofTheorem 1.4.2 As to any , the estimatorsatisfiesAs to the exponential population, people indicate a great interest in the parameterλas well.λis the very scale parameter. It appears as the rate of failure or of loss in thepractical application, and thus its various estimators come into being. In the case thatthe loss function is symmetric entropy loss function:, or q-symmetric entropy loss function , there are some outcomes[1],[2] about the estimation,yet they are all about only one exponential population with a contribution density asexpression(0.1). However, if we simultaneously estimate m rates of failure or of loss in thepractical application, m exponential populations should be brought in. We can usuallymeet with the cases that the ratios from the second to the m-th is directly proportionalto the first one with ratios known. In these cases, we need only estimate the first rate offailure or of loss. Now we introduce m exponential populations, the i-th of which has adistribution density as f_i(x): whereλis the parameter to be estimated, andαi is a plus known. We do our job inChapter 2 on the estimation ofλ, which concerns the m exponential populations intro-duced, under q-symmetric entropy loss function.At the beginning of the first section of this chapter, we list the conclusions inreference[1], which is about the the estimation ofλbased on a simple sample X =(X₁,X₂,···,X_n) in the case of only one exponential population. The first of them is asthe follows:If the prior distribution ofλisΓ(α,β), its unique Bayes estimator in the sense ofalmost surely equality iswhereThen, we continue to discuss a species of estimators with the form ofwhich includes all of the Bayes estimators ofλ, and obtain one of the Minimax estimatorsofλfrom the species. We denote The following is the conclusionsconcerned we list in the fist section:Theorem 2.1.2[1] In the case that the estimator admissible.Theorem 2.1.3[1] In the case that the estimator admissible.Theorem 2.1.4[1] In the case that the estimator admissible.Theorem 2.1.5[1] In the case that the estimator admissible.Theorem2.1.6[1] Under q-symmetric loss function, the estimator is among the Minimax estimators ofλ.Whereafter, in the end of the first section we study on the properties of strongconsistency and asymptotic normality of the Bayes estimators ofλ:And then we deduce the following conclusions:Theorem 2.1.7 As any the estimatoris the strongly consistent estimator ofTheorem 2.1.8 As to any and the estimatorofsatisfiesAt the beginning of the second section of the chapter, we illuminate that the esti-mator ofλin the case of m exponential population in the basis of m simple sampleswhich are respectively from the m population, is equivalent to that in the case of oneexponential population on the basis of a simple sampleTherefore, we easily derive the conclusions which is parallel to those in the first section: In the case of m exponential populations, with a prior distribution ofΓ(α,β),λhas a unique Bayes estimator as follows in the sense of almost surely equality:The following conclusion could be easily obtain:Theorem 2.2.2 In the case that , the estimator admissible.Theorem 2.2.3 In the case that , the estimator admissible.Theorem 2.2.4 In the case that , the estimator admissible.Theorem 2.2.5 In the case that , the estimator admissible.Theorem 2.2.6 In the case that , the estimator admissible.is among the Minimax estimators ofλ. Theorem 2.2.7 As to anyα∈(-∞,+∞) andβ∈(?∞,+∞), the estimatoris a strongly consistent estimator ofTheorem 2.2.8 As to any and the estimator ofsatisfies...