當給定一組來自連續型分布的隨機觀測值，傳統皮爾遜(Pearson)之適合度檢定(goodness-of-fit test)程序，係將觀測值加以分組，然後再計算卡方值 (chi-square statistcis)來進行檢定。然而，不同分組起始點(cell origin)的選取，極可能導致不同的檢定結果。並且，把連續型資料加以分組，也會導致的母體分布訊息及檢定力(power)有所損失。Wu and Deng(2010)提出一改良檢定方法，係計算L個不同起始點之K個區間（分組）的卡方檢定量，並以其平均值做為檢定統計量，我們稱此一統計量為移動平均式卡方檢定統計量(averaged shifted chi-square test statistics)。

　　本文主要目的在若干小樣本及L,K下，建立移動平均式卡方統計量的確切分佈，並與其漸近分佈比較兩者的機率差距程度，另外，我們也發現，將統計量進行微調後，其漸近分佈所求算出的機率值，能夠更進一步的接近確切分佈的機率值。

The classical Pearson’s procedure for testing whether a random sample has been drawn from a continuous distribution is based on the ‘difference’ of the observed cell counts and their model based counterpart. The test statistics known as the chi-square statistics can be very sensitive to the choice of cell origin. Different choice of cell origin may lead to different result of goodness of fits test. Worse still is that test based on grouping of data is often expected to be less powerful. To cope with the above two problems, Wu and Deng (2010) proposed to repeatedly partition the sample space into k cells for L times to obtain L respective chi-square statistics, and use their average to serve as the test statistics. Call the resultant test statistics the averaged shifted chi-square statistics (ASCS).

   The purpose of this thesis is to derive the exact distributions of ASCS for some small values of (L,k) by exhaustive permutation of all possible values ASCS. By comparing the exact distributions with the limit distribution of ASCS, we find that the limit distribution approximates well its target exact counterpart. A simple method is proposed to improve the approximation of the exact distribution by the limit distribution. Simulation study reveals that the power of ASCS is less sensitive to the choice cell origin and that ASCS can be more powerful as the values of k increases.

目錄
第 1 章 緒論 ........................................... 1
1.1 無母數適合度檢定簡介 ............................... 1
1.2 傳統卡方適合度檢定所遭遇之困難 ..................... 2
1.3 研究目的 ........................................... 4
1.4 章節編排 ........................................... 4
第 2 章 移動平均式卡方檢定統計量的介紹 ................. 5
2.1 移動平均式卡方檢定統計量的概念 ..................... 5
2.2 移動平均式卡方檢定統計量 ........................... 7
2.3 移動平均式卡方統計量的漸近分佈 ..................... 9
2.3.1 等距區間切割 ..................................... 9
2.3.2 等距區間切割下漸近分佈的模擬研究 ................ 11
2.3.3 錨點變動下檢定力的比較 .......................... 15
2.3.4 不等距切割條件下的漸近分佈 ...................... 19
2.3.5 不等距切割條件下的漸近分佈模擬研究 .............. 20
2.3.6 與文獻的結果比較 ................................ 23
第 3 章 確切分佈的建構與漸近分佈的改良................. 29
3.1 確切分佈的建構 .................................... 29
3.2 在漸近分佈的模擬改良 .............................. 32
3.3 確切分佈與漸近分佈模擬比較 ........................ 33
第 4 章 結論 .......................................... 39
參考文獻 .............................................. 40

表目錄
表1 不同起始點下之卡方檢定的結果 ....................... 3
表2 n=20,L=3,k=4之下，移動平均卡方統計量的計算法 ....... 8
表3 若干L,k下不同顯著水準相應的臨界點表 ............... 14
表4 等距與不等距下，傳統卡方與移動平均式卡方
的檢定力比較 .......................................... 26
表5 在n=10,若干小L,k的確切、修正前後漸近分佈尾端機率比較 .................................................... 36 
表6 在n=15,若干小L,k的確切、修正前後漸近分佈尾端機率比較 .................................................... 37A
表7 在n=20,若干小L,k的確切、修正前後漸近分佈尾端機率比較 .................................................... 38 

圖目錄
圖1 傳統直方圖與移動平均直方圖 ......................... 6
圖2 等距切割下漸近分佈與經驗累積分佈比較圖 ............ 13
圖3 對立假設為Gamma (1.3,1.1) 之下，變動起始點t及L下的
檢定力比較圖 .......................................... 16 
圖4 對立假設為Gamma (1.3,0.5) 之下，變動起始點t及L下的
檢定力比較圖 .......................................... 17
圖5 對立假設為Gamma (1.5,0.6) 之下，變動起始點t及L下的
檢定力比較圖 .......................................... 18
圖6 不等距切割區間下的漸近分佈與經驗累積分佈比較圖 .... 22
圖7 確切分佈流程圖 .................................... 31
圖8 n=15,若干L,k下的漸近、確切、經驗累積分佈比較圖 .... 34 
圖9 n=20,若干L,k下的漸近、確切、經驗累積分佈比較圖 .... 35

參考文獻
Anderson T.W. and Darling D.A.,1952 “Asymptotic theory of certain goodness-of-fit criteria based on stochastic processes” Annals of Mathematical Statistics, 23,193-212
Birnbaum Z.W. and Tingey F.H., 1951“One-Sided Confidence Contours for Probability Distribution Functions” Ann. Math. Volume 22,Number 4,592-596
D'Agostino, Michael A. Stephens, 1986 “Goodness-of-fit techniques ” New York : Marcel Dekker
Boero G., Smith J. and Wallis K. F.,2005“Sensitivity of the chi-squared goodness-of-fit test to the partitioning of data” Econometric Reviews, Volume 23, Issue 4,341-370
Gumbel E.J., 1943 “On the Reliability of the Classical Chi-Square Test ” The Annals of Mathematical Statistics, 14, No. 3 , 253-263
Henry C. and Thode. Jr, 2002 “Testing for normality ” New York : Marcel Dekker
Imhof J. P.,1961 “Computing the distribution of quadratic forms in normal variables” Biometrika, 48, No. 3/4., 419-426
Jones M.C., Samiuddin M., Al-Harbey A.H. and Maatouk T.A.H.,1998 “The Edge 
Frequency Ploygon” Biomerika vol.85,No1,235-239
Marsaglia G., Marsaglia J.C.W., 2004 “Evaluating the Anderson-Darling distribution”Journal of Statistical Software 9, 1-5
Scott D.W., 1985 “Average shifted histogram: Effective nonparametric density estimators in several dimensions ” Annals of Statistics 13:1024–1040
Scott D.W., 1992 “Multivariate density estimation: theory, practice, and visualization” New York: Wiley
Simonoff J.S. and Udina F.,1995“Measuring the Stability of Histogram Appearance when the Anchor Position is Changes” Economics Working Papers 133, Depeartment of Economics and Business
Stephens M.A., 1974 “EDF Statistics for Goodness of Fit and Some Comparisons ”Journal of the American Statistical Association, 69, No. 347, 730-737
Viollaz A.J. 1986 “On the Reliability of the Chi-Square Test ” Metrika, 33, 135-142
Wu.J. S. and Deng W. S., 2010 “Average Shifted Chi-Square test” working paper