The Statistical Inferences Of Sample With Heteroscedasticity

In the statistical inferences , the sample X₁,X₂...X_n are usually supposed to be i.i.d. But in some actual situations,identical distributions can't be contented.For example: in the pig breeding experiment,the piglets' weight is an important index of the parent's reproducting natraul .Because of the polyembryony of the pigs' reproduction, people often make the statistcal inferences by taking a small sample composed of a nest piglets' average weight, if there are n nest piglets in the same parent's,and individual weight obeys N(μ,σ²), and each individual are mutually independent,then, the n_i piglets' from the ith nest average weight obeys ,i = 1,2...n.Therefore, X₁,X₂...X_n are heteroscedastic. In fact,the sample with heteroscedasticity exist generally in some actual situations.Therefore,the statistcal references of sample with heteroscedasticity are very important in the actuality.This paper mainly studies the statistical renfernces of sample with heteroscedasticity.Let X₁, X₂...X_n are the sample with heteroscedasticity ,and mutually independent, X_i N(μ,σ_i²),i = 1,2...n. there already had many achievements about the statistical inferences of sample with heteroscedasticity, In this paper, I will studies this questions as follows:(1) For the sample with herteroscedasticity come from the one-dimentional normal distributed group, we will give the weighted estmator for the means,and prove that the estimator is uniform and sufficient.(2) For the sample with herteroscedasticity come from the multi-dimentional normal distributed group, we will give the best linear unbiased estimators in the R(A)-Criterion and R₂-Criterion that are weighted means estimator, and the estimator are uniform.(3) For two samples with herteroscedasticity come from the multi-dimentional normal distributed group: Xi,X2.-.Xni,Yi,Y2...Yn2, mutually independent, Xt N((xx,%),Yj Ninvfyw > 0,^ > 0,t = l,2...nltj = l,2...n2) Eti^ = l,^?iiw'- = 1. We will give the T-statistics for weighted means .If :/xx = /iy . Then: T F(m, m + n2 - m - 1).