在同時面對多個假設檢定時，犯型I誤差的機率往往成為要解決的問題。在典型的多重比較方法MCP (multiple comparison procedures)中，控制整體型I誤機率(familywise error rate，簡稱FWER)的方法，也就是使每個比較檢定給予相同犯型I誤差的機率(significance level)為常見的解決辦法之一。另一解決此問題之方向為Benjamini和Hochberg (1995)提出之控制錯誤拒絕率(false discovery rate，簡稱FDR)方法，但不論是FWER控制方法或FDR控制方法，皆須先估計虛無假設為真的個數。
由於最大概似估計量具有漸進不偏、漸進常態等良好之性質，本文針對估計虛無假設為真的個數，提出近似上限及最大概似估計方法，前者只需透過事前已知的檢定數及犯錯機率，即可得一近似上限；最大概似估計方法有別於過去以斜率和檢定統計量的觀點，是透過最大化概似函數以求得所需之估計量。此外，經由統計模擬和文獻上不同的方法以根號均方差(root mean square error) 作比較，模擬結果顯示在檢定個數較大時，本研究方法有最小的根號均方差，可以較準確地估計虛無假設為真的個數。

When we conduct multiple testing, the probability of committing type I error tends to become a problem to be solved. In typical multiple comparison procedures, the control of familywise error rate (FWER) method, which given each comparison test the same probability of type I error, is a common solution. Another solution to this problem is FDR-controlled procedure, which is proposed by Benjamini and Hochberg in 1995. Whatever FWER-controlled procedure or FDR-controlled procedure we choose, to estimate the number of true null hypotheses is the first thing needed to do.
Maximum likelihood estimation has some good properties such as asymptotically unbiased and asymptotic normality. For the problem of estimating the number of true null hypotheses, approximate upper bound and maximum likelihood estimation (MLE) are presented in this thesis. The former is obtained by the number of test and error rates; MLE is different from the past methods in the literatures, which is estimated by maximizing the likelihood function. In addition, we compare with different methods by root mean square error in statistical simulation. Simulation results show that when the number of test is large, the proposed method has the smallest root mean squared error. That is, MLE estimate the number of true null hypotheses more accurately when the number of test is large.

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