在完整資料下，兩組母體中位數之檢定已有不少學者提出無母數的檢定方法，例如：Mann-Whitney檢定、Mood’s median檢定。然而，這些方法的基本假設為兩母體分布之形狀需相同，如：變異數需相同。實際上，兩母體的變異數未必全然相同，所以Yu et al. (2011)以經驗概似比之檢定方法來檢定兩母體中位數的相等性。但是這些經驗概似比之檢定方法，僅針對沒有重複觀測值的資料類型進行推論，因此，本論文將其中兩個經驗概似比檢定的方法推廣至有重複觀測值的資料類型。此外，在針對Yu et al. (2011)的經驗概似比檢定方法(ELRII)進行模擬時，發現其型I誤差率達0.2，所以本論文修正這個經驗概似比檢定方法。除此之外，本論文也依據Tsai et al. (2015)和Lai (2014)提供的樣本中位數之變異數的估計量，進而建構Wald檢定統計量。最後，本論文的模擬結果顯示在小樣本時，以ELRII的表現為最佳，但在大樣本時，ELRII和兩個Wald檢定之檢定力表現相當，由於在型I誤差率ELRII之表現較為穩定，所以依舊推薦使用ELRII。

Several tests have been developed for comparing the medians of two populations with complete data, such as Mann-Whitney test and Mood’s median test. However, those tests require that the shape of the two population distributions be the same in order to derive the asymptotic distribution of the test statistics under the null assumption. Therefore, Yu et al. (2011) propose empirical likelihood ratio tests to avoid the above limitation. Their procedure is developed for data with no ties; we modify their procedure for data with ties in this thesis. Moreover, the type I error rates of their procedure (ELR II) appear to be larger thn expected under our simulation study. Therefore, we examine their procedure in detail and discover two likely typos in the original Yu et al. (2011) paper. Through correcting the typos, we are able to control the type I error rates at desired levels.

It is natural to use the Wald test for comparing two population medians. To avoid estimating nonparametrically the unknown probability density function, the variance estimators of sample median proposed by Tsai et al. (2015) and Lai (2014), respectively, are employed to construct two Wald tests. We use a simulation study to compare the type I error rate and the power of ELR II and the two Wald tests. The performance of ELR II is the best under small sample sizes. The performance of ELR II and the two Wald tests are similar for large sample sizes, but type I error rate of ELR II is more stable than those of the two Wald tests. Thus, we recommend to use ELR II for comparing two population medians.

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