最高密度顯著性檢定

Abstract

顯著性檢定是一種藉由計算p值的方法，用於衡量是否違反虛無假設的統計證據，傳統的顯著性檢定選擇一個統計量T = t(X)，同時決定一個極端的集合，此集合包含比觀測值 t(x0)極端或相當的所有點。但是，這個方法有可能無法找到一個合適的統計量，或者不存在一個普遍性的最佳性質來支持既存的顯著性檢定。因此，我們提出一個新的顯著性檢定，設定極端集合包含所有發生機率比觀測值x0機率小的點，稱為最高密度顯著性(HDS)檢定。此方法應用到較小的機率顯示存在更強的證據否定虛無假設的概念，且將一個樣本X藉由機率比分為極端與非極端的兩個集合。在相同的p值檢定中，我們發現HDS檢定的非極端集合體積最小，此為一最佳性質。我們更進一步延伸HDS檢定來建立控制圖，同時監控所有的參數，並且能精準的達到第一類誤差的機率。藉由監控樣本點的機率來辨識是否受到控制，不像傳統的控制圖是依據樣本平均和全距來監控。

The significance test is a method for measuring statistical evidence against null hypothesis H0 by computing p-value. The classical significance test chooses a test statistic T = t(X) and determines the extreme set representing the sample set with values t(x) greater than or equal to t(x0), where x0 is the observed sample. It may be difficult to choose a suitable test statistic for the test, or there is no generally accepted optimal theory to support the existed significance tests. Now, we propose a new significance test, called the highest density significance (HDS) test, setting extreme set including those sample points with probabilities less than or equal to it of x0. It applies the concept that the smaller probability of an observation X = x0 reveals stronger evidence of departure from H0. This test virtually classifies the sample space of random sample X into extreme set and the non-extreme set through a concept of probability ratio. We also show that this test shares an optimal property for that it has smallest volume among the class of non-extreme sets of significance tests with the same p-value. Further, we extend HDS test to set up a control chart which can monitor all the parameters simultaneously and the probability of type I error is precisely attained. Unlike the classical control charts that track statistics such as sample mean or sample range R, it is tracking the density value of the sample point to classify if it is in control.