頻率移動平均卡方統計量

Moving Average Frequency Chi-square Statistics

蕭任嫻(Jen-Hsien Hsiao)

指導教授 ： 伍志祥

淡江大學/理學院/中等學校教師在職進修數學教學碩士學位班/碩士(2013年)

https://doi.org/10.6846/TKU.2013.00113

摘要

給定一組連續型分布的隨機樣本，傳統的卡方適合度檢定會因分組起始點選取的不同，導致不同的檢定結果，Wu 和Deng（2010）提出移動平均卡方檢定統計量，來改善個問題。在模擬研究時，發現在對立假設的某一分布下，相鄰樣本數的檢定力會產生不小差異，這表示卡方適合度檢定對樣本數敏感。
本論文提出頻率移動平均卡方統計量（moving average frequency chi-square statis- tics），是一般化Wu 和Deng（2010）的方法。首先，將[0,1]切割成m個區間，並計算每一區間擁有的樣本頻率，再取相鄰的l個區間的樣本頻率值，計算l階頻率移動平均值，再以此l階移動平均值為基礎，建立卡方統計量，並稱此為頻率移動平均卡方統計量（MAFCS）。我們可以藉由U統計量的理論，證明MAFCS的漸進分布會趨近於有限個自由度1的卡方變數的線性組合。利用模擬數據分析可知，MAFCS可改善檢定力對樣本數的敏感，另外對於震盪頻率較高的分布函數，MAFCS可提供比尼曼平滑檢定與Anderson Darling檢定更高的檢定力。

關鍵字

頻率移動平均卡方統計量 ； 適合度檢定

並列摘要

Given a set of observations from a continuous distribution, consider the problem of testing whether the sample has been drawn from a population with a specified probability density based on grouping of data. The chi-square test would be very sensitive to the choice of anchor (cell origin) and lead to different test results of power, between adjacent sample sizes, such not as a reference to each other. Therefore, in this presentation, it is the idea of moving average frequency that gives rise to generalize the averaged of shifted chi-squared test, proposed by Wu and Deng(2010). Computing moving average frequency values and use these values to construct chi-square statistics. Call the proposed test statistics moving average frequency chi-square statistics (MAFCS). By the theory of U-statistics, we prove that the proposed MAFCS is asymptotically distributed as a finite linear combination of chi-square variables of degree 1. The simulated power comparisons show that, MAFCS can improve the pro- blem of reference the results between adjacent sample sizes and lead to better gains than Neyman Smooth tests and Anderson-Darling tests in power.

並列關鍵字

moving average frequency chi-square statistics ； goodness-of-fit test ； overlapping cells

參考文獻

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