有限樣本母體分配假設的訊息矩陣檢定

INFORMATION MATRIX TESTS FOR POPULATION DISTRIBUTION HYPOTHESIS IN FINITE SAMPLES

母體分配的假設是個重要的議題。如果當初關於母體分配的假設不恰當，後續的統計推論都將失真。本研究探討了White（1982）母體分配假設的訊息矩陣檢定法（information matrix test）及其相關的檢定法。鑑於該檢定法必須計算logdensity之第三階偏導數以求得共變異矩陣，因此推演困難且繁雜，並不實用。Chesher（1983）與Lancaster（1984）提出人工迴歸式的簡單計算公式，不需計算▽D(θ)，就能估算出White訊息矩陣統計檢定式等於樣本大小n乘以判定係數R2。由於該人工迴歸式不適用於▽D(θ)為0的情境，因此我們提出了適用於▽D(θ)為0時，和Lancaster計算式不同的統計檢定式。由於訊息矩陣檢定法，牽涉到許多估計式，在尚無嚴謹理論佐證下，我們藉由模擬分析來評斷各檢定式之第一型誤差，結果顯示Lancaster人工迴歸檢定法傾向於過度拒絕虛無假設，White檢定法結果是三者中最令人滿意的，我們的檢定法結果介於兩者之間，這說明了我們的檢定法簡單有效。

This study attempts to explore the issues in hypothesis testing of population distributions, based on Whites' information matrix test (White, 1982). We point out that White's ▽D(θ) method requires the computation of the third derivatives of log-density for finding the covariance matrix, which is very labor-intensive and impractical. Chesher (1983) and Lancaster (1984) developed a simpler method of artificial regression where the computation is no longer needed. It is found that White's w is equal to sample size n multiplied by the coefficient of determination R2. However, their method is improper when ▽D(θ) = 0. Accordingly, we propose another estimator to correct the errors in the artificial regression. Without strong theoretical evidence, these three methods are compared through a simulation study in terms of type I error rates. The results show that the artificial regression method tends to over-reject the null hypothesis, White's method yields very satisfactory results. Our method is between these two methods in performance, indicating that our method is simple and effective.