The Parameter Estimation Under The Compound LINEX Symmetric Loss Function

The symmetric loss function is of great importance in the theory of statistical decision,such as squared loss function, which reflects the fact that if an actionδis close toθ, thenthe decisionδis reasonable and little loss is incurred. Ifδis far fromθ, then a large loss isincurred and thereforeδis not good. In fact, the admissibility of the estimator may dependquite sensitively on features of the loss function. So it is important to do more study of theproperties of alternative estimators relative to other types of symmetric loss function. Thepresent paper deals with the Bayes estimation for the mean parameter of normal distributionand the parameter of exponential distribution under Compound Linex symmetric lossfunctions with the form of L(θ,δ)=e^{-α(θ-δ)}+e^{α(θ-δ)}-2.The first and second chapters of the article give a brief account of the back ground of thetheory of statistical decision and Bayes decision, the concept of admissibility andinadmissibility, and so on.Chapter 3, it deals with the Bayes estimation for the mean parameterθof normaldistribution under Compound Linex symmetric loss functions with the form ofL(θ,δ)=e^{-α(θ-δ)}+e^{α(θ-δ)}-2. Assume that the prior pdf forθisN(μ,σ²), that is a normal pdfwithμmean and varianceσ². Then the Bayes estimation contains the form:δ=cT(X)+d,(T(X)=(?),c,d are parameters). Under four different assumptions of the parameters c and d,the admissibility and inadmissibility of the estimations are discussed.Chapter 4, the Bayes estimation of the exponential distribution under Compound LINEXsymmetric loss functions and the problem of its admissibility and inadmissibility arediscussed.Conclusion of this work is done to conclude the estimation method, moreover we discussthe feasibility and what we have to improve and explored in deep research.