在無母數統計的領域中,可以找到許多關於對稱性問題的文獻,大部分對於對稱性問題的檢定都是在對稱中心已知的情況下所討論,其中一個例子就是1990年McWilliams以run statistic為基礎提出的檢定統計量 R*,而1996年Modarres和Gastwirth將McWilliams的方法做了修正,提出檢定統計量 Mp,1999年李俊達和鄭惟厚以統計量 Mp 的觀點為基礎,提出檢定統計量 Ck。實際上大部分的情形可能並不知道對稱中心為何,所以2004年鄒瑞穎和鄭惟厚針對對稱中心未知的情況做了討論,她們以樣本中位數來當做對稱中心的估計,但在某些分布下中位數不見得會是適當的估計,所以我們現在考慮引進Hogg在1967年提出的因應統計量概念,首先先從樣本獲得一些初步的訊息,再依訊息選擇適當的方式來做後續的統計推論。
本文就是探討在對稱中心未知的情形下,引進Randles, Ramberg and Hogg(1973)所提出的因應方法(adaptive method),先將樣本分成輕尾(light-tailed),中尾(medium-tailed),重尾(heavy-tailed)三類,再依各類的特性選用適合的統計量來當對稱中心的估計,接著定義一個概念跟 Ck 相同的統計量用來檢定對稱性的問題,並將結果和用中位數做估計的結果做比較。

Most of nonparametric methods for testing symmetry focused on the case of a known center. For example, McWilliams(1990) presented a test statistic R* based on a run statistic. Modarres and Gastwirth(1996) presented a test statistic Mp by using Wilcoxon scores to weight the runs. Chun-ta Li and Wei-hou Cheng(1999) presented a new test statistic Ck which is very easy to apply. Jui-yin Tsou and Wei-hou Cheng(2004) considered the situation of unknown center, they estimated the unknown center by sample median before applying the same idea as Ck. Since median is not always a suitable estimate of the center of a distribution, we consider an adaptive procedure which is presented by Hogg(1967).
    In this paper, we consider the situation when the center is unknown, using an adaptive method which is presented by Randles, Ramberg and Hogg(1973). We first classify the sample as light-tailed, medium-tailed and heavy-tailed, then estimate the center by using suitable statistic and construct a test statistic similar to the test statistic Ck.

1 前言1
2 文獻回顧3
2.1 Thomas P. McWilliams 在1990年所提出的連串檢定. 3
2.2 R. Modarres 和J. L. Gastwirth 在1996年所提出的
修正式連串檢定. . . . . . . . . . . . . . . . . . . . . . . 5
2.3 李俊達和鄭惟厚在1999年所提出的簡易對稱檢定. . . . . 6
2.4 Robert V. Hogg 在1967年所提出的因應統計量. . . . 8
3 對稱性問題的討論10
4 蒙地卡羅模擬13
5 結論17
參考文獻19

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