兩個卜瓦松分佈的統計檢定方法

Abstract

卜瓦松分佈被普遍運用在許多流行病學、工業品質管制問題上。Ng與Tang(2005)曾介紹過ㄧ個實際範例，研究人員欲比較兩個群體中某個特定疾病的發生率，則研究人員以卜瓦松分布來描述群體中疾病發生個數的分布狀態，而研究目的在比較此兩個群體的平均發病率。已知當樣本數或平均發生個數夠大時，則統計人員得運用常態分布近似在相關的漸近統計檢定方法的發展上。但上述條件不滿足時，則應運用確實統計檢定方法(exact testing procedure)以獲得適當的統計結論。本研究的目的為在比較兩個卜瓦松分佈的問題中，發展確實統計檢定方法。此時虛無假設參數空間將牽涉干擾參數，而干擾參數的出現將大幅增加確實統計檢定方法的計算複雜度。我們將研究相關的理論以減少檢定方法的計算量。我們將考慮受限最大概似估計方法在檢定統計量上，另一方面，由Berger與Boo (1994)提出的信賴域p-value也將被研究，其中將牽涉在受限參數空間中的信賴域。我們將研究所提出的統計檢定方法的有效性性質，並將透過電腦模擬來驗證這些方法。

The Poisson distribution is suitable in variety fields such as biology, quality control, and so on. For example, one is interested in comparing the incidence rates of some disease in different groups (Ng and Tang, 2005). The number of cases in a given person-years at risk is modeled by a Poisson distribution and the mean difference is tested. When the sample sizes or the mean rates are sufficiently large, the normal approximation is applied for constructing asymptotic tests. On the other hand, when the approximation fails, an exact test is more adequate. This study will develop exact testing procedures for comparing two Poisson means. However, in this problem nuisance parameters present in the null exact sampling distribution of the test statistic. It greatly increases the computational difficulty in applications. One important aim of this study is to construct exact testing procedures with computational efficiency. Some restricted estimates of the nuisance parameters will be employed in calculation of the p-value. On the other hand, a confidence-interval p-value proposed by Berger and Boo (1994) will be also considered. The validity of the proposed methods will be investigated theoretically and empirically.