[[alternative]]Aaveraged Shifted Chi-Square Test of Bivariate Goodness of Fits

計畫編號：NSC101-2118-M032-001 研究期間：201208~201307 研究經費：447,000[[abstract]]當給定來自於一連續母體的二維樣本1 1 2 2 ( , )( , ), , ( , ) n n X Y X Y X Y ，實務上我們常 利用分組資料(grouped)來檢定此一樣本是否來自於某特定二維連續分布。最古老 且最為人熟知的檢定程序即是皮爾森卡方檢定 (Pearson’s Chi-square test)。然而， 此一檢定程序有兩大缺點：第一，它對於不同的樣本空間之分割(partition)及分組 起始點(cell origin)極為敏感，分組起始點的些微變化極可能導致檢定統計量的大 幅變化。第二，由於此一檢定程序係是將連續型資料轉換成為分組資料，因此會 損失資料所包含的訊息，並導致檢定力的降低。本計劃中，我們將提出一個新的 檢定程序來改善上述兩個缺點，這個新的檢定程序的做法是先將樣本空間做2 次的不同分割，再對於每一個個別的分割，分別計算出對應的卡方統計量，最後 再把這2 個卡方統計量加以平均來做為檢定統計量，我們稱此檢定程序為移動 平均式二維卡方檢定(averaged shifted bivariate chi-square test)。我們將推導此一檢 定統計量的漸近分布，並利用 Viollaz(1986) 和 Wu and Deng(2012) 的方法來分 析它是否能降低卡方值對分組起始點的敏感程度。最後，本計劃將進行一系列的 模擬研究，來檢視此一新程序的檢定力。由於此一檢定方法係利用Scott(1985) 及Wu and Deng(2012) 的想法所提出，因此我們可以期待它能改善傳統卡方檢定 對分組起始點的敏感程度，並享有較高的檢定力。[[abstract]]Given a random sample 1 1 2 2 ( , )( , ), , ( , ) n n X Y X Y X Y from a continuous distribution, it is a common practice to test whether the sample has been drawn from a specified continuous joint density based on grouping of data. The oldest and best known procedure that serves this purpose is Pearson’s chi-square test. However, the classical procedure suffers from the problem of being sensitive to the placement of cells used to group the data. Worse still is that the grouping of continuous data into non-overlapping cells often leads to the loss of information and thus the power performance. In this project, we will propose a procedure to cope with the above two drawbacks of the classical test. The remedy proposed in this project is to repeatedly partition the sample space for, say, 2 times with respect to 2 shifted cell origins, to obtain 2 shifted hi-square statistics. We defined our test statistics as the averaged of the shifted chi-square statistics calculated from each partition. We will endeavor to obtain the limit distribution of our proposed test statistics and investigate its sensitivity to the choice of cell origins based on the theoretical method of Viollaz (1986) and Wu and Deng (2012). A Monte Carlo study will be carried out to gains some insight into the power performance of our procedure. The remedy can be viewed as an extension of the strategy of Scott (1985) and Wu and Deng (2012) that serves in the case of univariate goodness of fits test. Thus, our proposed procedure can be expected to be less sensitive to the choice of cell origins as well as to have noticeable gains in power.[[sponsorship]]行政院國家科學委員