随机估计和VDR检验

众所周知统计推断有三种理论:普遍承认的Neyman理论(频率学派),Bayes推断和信仰推断(Fiducial)。Bayes推断基于后验分布,由先验分布和样本分布求得。信仰推断是基于信仰分布(Confidence Distribution,简称CD),直接利用样本求得。两者推断方式一致,都是用分布函数作推断,称为分布推断。从分析传统的参数估计、假设检验特性来看,经典统计推断也可以视为分布推断。通常将置信上限看做置信度的函数。其反函数,即置信度是置信上界的函数,恰是分布函数,该分布恰是近年来引起许多学者兴趣的CD。在本文中,基于随机化估计(其分布是一CD)的概率密度函数,提出VDR检验。常见正态分布期望或方差的检验,多元正态分布期望的Hoteling检验等是其特例。VDR(vertical density representation)检验适合于多元分布参数检验,实现了非正态的多元线性变换分布族的参数检验。VDR构造的参数的置信域有最小Lebesgue测度。

Random Estimate and VDR Test

There are three way of statistical inference,Neyman’s theory,Bayes’s throry and Fiducial inference.R.A.Fisher proposed fiducial probability distribution as his alternative to Bayesian posterior prabability distribution.Posterior distribution are determined by prior distribytion and sample,while fiducial distribution are obtained derictly from sample.They infer parameters by distributions and Neyman’s infrence of paprameters consider as inference by CD(Confidence Distribution).If consider the reasoning leading to fidicial probability density,then there is little wonder that Fisher caused such puzzles with his novel idea.We use random vector(randomized estimator) defined by using pivotal quantity to estimate parameters and the prabability density of randomized estimator is just the fiducial density and it avoid Fisher puzzles.We proposed VDR(Vetical Density Representation) test by the density and VDR theory.The classical tests will be derived when appliy VDR test to certain case,for example t-test, Hoteling test on multinormal mean vextor etc.and some new test are also derived,for example,test covariance matrix of multinormal normal distributions.The confidence rigion constructed by VDR has the smmalest Lebesgue measure,it will be called minimum confidence rigion with Lebesgue measure.