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.