Minimax invariant prediction regions
Minimax invariant prediction regions
复制标题
最小最大不变预测区域
DOI：
10.1007/bf02911623
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发表时间：
1968
影响因子：
1
通讯作者：
G. Ishii
中科院分区：
文献类型：
--
作者：
G. Ishii
观测~n个独立随机变量X1，X2，. . .，X~，作为关于n个未来独立随机变量Y~，Y，. .，Y~，我们感兴趣的是变量Y=（Y，Y ~ 2，.如果可以观察到X~和Y ~的分布律具有相同的参数0，则Y~）将落入。这样的区域被称为预测区域。表示X=（X，X2，...，X，~），我们将在Y的值域空间中寻找一个集合Sr，使得对于所有的~ 0，Po{Y~ S~} >_1-~，并且在某种意义上Sx尽可能小。当n = 1时，预测区域与Fraser和Guttman [2]的E-期望容差区域一致。石井和Kudo [4]首次对许多未来随机变量的预测区域进行了处理，其中潜在的分布规律被限制为正态情况。极大极小不变原理在文献[4]中起了重要作用。这里的目的是从更一般情况下的相同观点来处理预测区域。我们在这里对该区域采用的可取性度量是该区域的体积（或面积）的倒数。该测度类似于最小长度置信域的测度，并根据作者的观点，基于人类预测行为的倾向性。每个预测区域都涉及两个风险，即该区域的平均概率含量和平均体积。我们的目标区域是极大极小相对于平均体积内的类的预测区域的平均概率内容不小于一个预先指定的常数1-s（水平1-e）。第二节介绍了文[4]中的一些结果。第三节讨论了非正态分布下极大极小不变预测域的推导。对于n t，它们中的一些与Guttman [3]中的容忍区域相吻合.多元线性模型在第4节中讨论。其中，《易经》与《易经》是相似的，《易经》与《易经》是相似的。当n= 1时，多元正态情形的结果与Fraser和Guttman [2]的结果不一致
Observing ~n independent random variables X1, X2 , . . . , X~, as a statistical inference about n future independent random variables Y~, Y , . . , Y~, we have an interest in a region which the variable Y= (Y, Y2, . ' , Y~) would have fallen in if it could be observed where the distributive laws of X~ and Yj have the same parameter 0. Such a region is called a prediction region. Denoting X=(X, X2,..., X,~), we shall search for a set Sr in the range space of Y such that Po{Y~ S~} >__1--~ for all # ~ 0 and in some sense Sx is as small as possible. If n = 1, the prediction regions coincide with E-expectation tolerance regions of Fraser and Guttman [2]. For many future random variables the prediction regions were treated by Ishii and Kudo [4] for the first time where the underlying distribution law was restricted to normal case. A minimax invariant principle played an important role in [4]. The purpose here is to treat the prediction regions from the same point of view for more general cases. The measure of desirability that we adopt here for the region is the reciprocal of the volume (or area) of the region. This measure is analogous to that of confidence regions of the smallest length, and based on the tendency of human being in prediction behavior according to the author's opinion. There are concerned two risks for each prediction region, the average probability content and the average volume of the region. Our aimed region is minimax relative to the average volume within the class of prediction regions with the average probability content not less than a preassigned constant 1--s (level l e ) . In section 2 some results in [4] are described. Section 3 is devoted to the derivation of the minimax invariant prediction region in some non-normal distributions. For n t , some of them coincide with the torelance regions in Guttman [3]. Multivariate linear model is treated in section 4. The method employed there is similar to that of Stein [7] and Wijsman [8]. For n=l, the result for multivariate normal case does not coincide with that of Fraser and Guttman [2] which is caused