Parameter Estimation Under Asymmetric Loss

Statistical decision plays a very important role in mathematical statistics. Whatis different from Classical statistics, Decision theory introduces the loss function inthe statistical analysis to make quantification analysis in different decisions whichcause the losses. Among them, Parameter estimation is a very important direction. Inthe previous studies, because the symmetric loss function is very convenient forcalculation and discussion, the statisticians have done a lot of work for the parameterestimation under the loss and obtained a lot of good results. However, with theapplication of this theory more widely, the symmetric loss function cannot trulyinterpret all the problems in the practical application. In many cases, overestimatingthe parameter has a quite different impact from underestimating it. Obviously, theasymmetric loss function is more appropriate at this moment. Therefore, the researchon parameter estimation problem under the asymmetric loss makes a lot of sense.The point estimation and the interval estimation are very important of all theresearch directions of the parameter estimation. This paper mainly takes these twoaspects into account, and it is divided into the following several parts.At first, it introduces the research background and the development situation ofthe statistical decision theory, the Bayes analysis and the asymmetric loss function inthe introduction. Secondly, in the second chapter it introduces some basic conceptsneeded in this article. The third chapter makes a summary of the point estimationunder the asymmetric loss, mainly provides some excellent results from thedomestic and overseas research in this area. The fourth chapter is the main part ofthis paper, it takes the interval estimation under the asymmetric loss intoconsideration. Linex loss function is a common asymmetric loss function. It hasmany good properties. In the past the studies of this area was mainly concentrated under this loss function. This paper mainly discusses the interval estimation under the Linex loss function too. First it simply lists the obtained conclusion. Let the prior distribution be N(0,k2), when σ2is known, the Minimax confidence interval estimation and the admissible condition of the normal mean6of the sample X～N(θ,σ2) is obtained under the Linex loss function. Then we generalize the prior distribution to the more general situation N(ξ,k2). By using the posterior distribution of θ directly, we provide the Minimax confidence interval estimation (δL*,δU*) of the linear function θ=aθ+b,a∈R+,b∈R of θ under the Linex loss function, and we also prove its admissibility. Finally, we can obtain that if we consider using the posterior distribution of θ, the result is as the same as the conclusion that directly use the posterior distribution of θ.