| Abstract: | 在一般用參數檢定(parametric tests)估計未知位置參數時,利用雙尾檢定找未知參數的接受域(acceptance region),就可以得到信賴區間(confidence interval),我們將這個概念用在從排列檢定(permutation tests)找信賴區間。對於一般的參數檢定問題,通常只要找到樞紐(pivot)統計量,再利用簡單的代數就能找到信賴區間,但是從排列檢定找信賴區間,狀況卻複雜許多。
於單一樣本的情況下,排列檢定是將觀測值減去原始假設下的參數之後,討論所有可能的正負號分配情況,得此假設下的排列分布,再根據此排列分布來判斷是否接受原始假設。不過每減去不同的參數值,所對應的排列分布也必須重新計算,這使得計算過程極為繁瑣。本文討論的是不同信心水準信賴區間所對應的條件,並試圖用較精簡的方式將這些條件表示出來以利應用。另外並提供了程式,在 n=7 的情況下可直接利用此程式找出信賴區間。
我們也利用找出的條件檢驗了來自標準常態分布、均勻分布及標準雙指數分布的樣本,檢測從何者抽出的樣本較容易滿足信心水準 90% 以上的信賴區間之條件。另外,若樣本來自連續對稱的單峰分布,在相同「位置間距」下,也討論了「不對稱位置」的區間和「對稱位置」的區間何者較長。
In general case of parametric tests,we can usually find the confidence interval of an unknown location parameter via the acceptance region of a two-tail test. In this paper,we use this same concept to find confidence interval based on permutation tests. But in parametric tests, the unknown parameter usually appears in the pivot and we can find the confidence interval via simple algebra. Yet in the case of permutation tests,the situation is much more complex.
To carry out a permutation test on a center of symmetry,we subtract the parameter under consideration from each of the observations to get the permutation distribution,and then decide whether to accept the null hypothesis based on the permutation distribution. However,the computing process is very tedious,because the permutation distribution has to be recomputed every time we subtract a different parameter. We discuss conditions under which there exist confidence intervals corresponding to different confidence coefficients. We also try to find ways to simplify these conditions to make applications easier. In the case of n=7 ,a program is supplied for finding confidence intervals.
We compared samples from standard normal,uniform and standard double exponential distribution to find out which one satisfies the conditions for existence of 90% (and above) confidence interval more often. We also compared the length of intervals whose end points have "asymmetric locations" with those that have "symmetric locations" and obtained a result for a special case. |
