移動平均皮爾森卡方檢定統計量的研究
學年 97
學期 1
出版（發表）日期 2009-01-01
作品名稱 移動平均皮爾森卡方檢定統計量的研究
作品名稱（其他語言） Average Shifted Pearson Chisqure Test
著者 伍志祥
單位 淡江大學數學學系
描述 計畫編號：NSC98-2118-M032-001 研究期間：200908~201007 研究經費：438,000
委託單位 行政院國家科學委員會
摘要 檢定虛無假設：樣本來自某一連續型分佈，是否成立，被稱為適合度檢定 (goodness-of-fit test)。文獻中有很多適合度檢定的方法，其中最被多數人所知道的是 Pearsonχ 2 檢定(Pearson’s chi-square test)。等距離分割的Person χ 2 統計量，會受起始點 選取的影響，產生檢定適當性的疑惑。若把Pearson χ 2 統計量視為起始點的函數，則為 隨機過程。Viollaz(1986)推導這隨機過程的自我相關函數(autocorrelation function)，發現 對任意給定的起始點，會存在另一起始點，使得這兩起始點的卡方統計量的相關係數 (correlation coefficient)小於0.5。這解釋Person χ 2 檢定會因起始點的因素造成適當性問 題的原因。文獻中有Hall(1985)的改良型 χ 2 統計量跟Viollaz(1986)的連續型 χ 2 統計量， 可以避免因起始點因素引起適當性的爭議。但其在虛無假設下的極限分布與由可數無窮 獨立 χ 2 隨機變數的線性組合而成的隨機變數的分布一樣，所以極限分布的分位數 (quantile)的計算頗為麻煩且費時，目前沒有分位數表可供使用者查詢。 這個計畫是想提出一種折衷型的改良式 χ 2 統計量，對起始點的選取不會敏感， 且其在虛無假設下的極限分布的分位數的計算簡單。我們採用Scott(1985)移動平均直方 圖(average shifted histogram)的想法，提出移動平均 χ 2 統計量(average shifted chi-square statistics)來減低起始點的影響。至於要選擇幾個Person χ 2 統計量來平均，可以透過自我 相關函數的研究來決定。 另外關於分位數計算的問題，我們已經試著推導某些設定下的移動平均 χ 2 統計量 在虛無假設下的極限分布，與由有限個獨立指數隨機變數的加權和(weighted sum)而成的 隨機變數的分布一樣。這種類型的分布，其分位數的計算比 χ 2 分布的分位數的計算容 易。有此認知，我們相信在其他設定下的移動平均 χ 2 統計量，其在虛無假設下的漸近 分布的分位數計算也不會複雜到難以計算。 Given a random sample of size n, consider the problem of testing the null hypothesis that the sample has been drawn from a continuous distribution. The oldest and best known goodness of fit test that serves this purpose is the Pearson chi-squared test (Pearson (1900)). The primary stage of the procedure also serves the construction of histogram. So, as might be suspected, the goodness of test does not yield unique result, since the value of test statistics depends on the number of class intervals and the choice of these intervals’ anchor position. Violaz(1986) shows that, given a fixed anchor position x, there exists another anchor point y such that the correlation coefficient of the two chi-square statistics using x and y, respectively, is smaller than 1/2. Hall (1985) and Violaz(1986) modify Pearson’s chi-square statistics to suppress the noise effect of anchor shifting. Unfortunately, the limiting distributions of their modified statistics involve countably infinite number of chi-square variables. The calculation of the limiting distributions’ quantile is quite computationally tedious and there is no table available for inferential purpose. Motivated by the idea of average shifted histogram (Scott (1985)), I shall propose a method to modify the chi-square statistics that is less sensitive to the choice of anchor point and computationally simple for calculation of its limiting distribution’s quantile. The new device is the average of, say, k chi-square statistics evaluated on different collection of class intervals with different anchor positions. Central to this project, I shall derive the asymptotic distribution of the modified test statistics. I shall focus on the case that the null distribution is uniform(0,1) , without loss of generality due to probability integral theorem. For the case of k=2 and class width=1/3, it has been known that the asymptotic distribution is the same as that of the sum of finite number of independent exponential variables. The asymptotic distributions for other cases of k and class width are believed to be ease to derive and the calculation of its quantile is believed to be computationally simple.
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