| Abstract: | 大部分的參數檢定與無母數檢定問題都存在一個相對應的排列檢定(permutation test)處理,不管樣本是連續、順序或是類別的型態都可以應用,且排列檢定有時還有相同或更好的檢力(power)。而Good[1]在書中敘述,一般常見的無母數檢定當中,有百分之九十九可以視為將原始資料由rank取代後的排列檢定。本篇論文討論的無母數檢定是參考Conover[2]一書,藉由排列檢定的特性(equally likely)來探討無母數檢定與排列檢定間的關係。我們的討論方式是想辦法產生一些機率相同的結果,再整合出原先無母數檢定的統計量各種可能值的機率,如此便可以找到對應的排列分布。
我們得到的結論為:在一般的無母數檢定中,如果檢定與binomial test相關,基本上都可以視作排列檢定的一種;與卡方檢定相關的檢定,基本上和p值已知或是未知有關,這個p值為原始假設下所要檢定的母體機率值;其他檢定如Kolmogorov goodness-of-fit test,由於統計量的值是根據empirical distribution function決定,而不論如何重新排列樣本,其順序統計量都不變,所以無法視為一種排列檢定。
總結來說,大部分的無母數檢定可以視為排列檢定,關鍵在於是否可以找到一個合適的排列,而不同的問題中,「排列」可能有不同的定義。
For most parametric and nonparametric tests，there is a permutation counterpart，and permutation test can be applied to all kinds of data，including continuous、ordered and categorical data，and the resulting permutation test is often as powerful as or more powerful than alternative approaches．Good[1] made a interesting statement in his book：ninety-nine percent of common nonparametric tests are permutation tests in which the original observations have been replaced by ranks，but he did not elaborate．We want to check out this statement．In this thesis，we discuss the connection between nonparametric tests and permutation tests by the property of permutation test (equally likely) and the nonparametric tests we consider are those in Conover''s book[2].
After comparing the two type of tests，we came to some conclusions．The tests based on binomial test can be regarded as some kind of permutation tests under certain definition of "permutaton"．If the tests are related to statistics，then whether they can be viewed as permutation tests depends on if the value of the probability，p，is known where p is the population parameter in the null hypothesis．For tests like Kolmogorov goodness-of-fit test，since the statistics is calculated based on empirical distribution function，hence no matter how we repermutate the observations，the order statistics is the same．This kind of tests can''t be regarded as permutation tests.
To sum up，we find that most of nonparametric tests can be regarded as permutation tests，and the key point is to find a appropriate definition of "permutation"． |
