A Comparison on Confidence Regions for the Mean and Variance of a Normal Distribution

常態分配下平均數與變異數信賴區域之比較

蔡旻曉(Min-Hsiao Tsai)

《統計與資訊評論》 17期 (2016/09) Pp. 69-82

摘要

Let X_1, X_2, ..., X_n be independent and identically distributed random variables from a normal population with unknown mean μ and variance φ. This study deals with the problem of constructing a 100(1 - α)% confidence region for μ and φ simultaneously. There are three likelihood-based confidence regions (say, LCR, WCR and SCR) can be obtained by inverting the acceptance regions of likelihood ratio test, Wald's test and Rao's score test, respectively. However, these results are only asymptotically correct. Let X and S^2 be the sample mean and sample variance, respectively, then by the facts that (i) T_1 = n(X - μ)^2/φ∼ X_1^2, (ii) T_2 = (n-1)S^2/φ∼ X_(n-1)^2 and (iii) T_1 and T_2 are independent, one can construct an exact 100(1-α)% joint confidence region (say, JCR) for (μ, φ). Moreover, the better-known Bonferroni confidence region method (BCR) for constructing simultaneous confidence intervals is also considered. In order to get an insight of which method is more suitable for the problem, we evaluate the coverage probabilities and coverage areas of these considered confidence regions by a Monte Carlo simulation. Simulation results show that the JCR method has more accurate coverage probability to the specific confidence level 1-α for a wider range of sample size, whereas the LCR (BCR) method tends to be somewhat lower (higher) when sample size is small to moderate. Moreover, the WCR method gives the smallest coverage area but it is the worst one in the sense of having lowest coverage probability. On the other hand, the SCR method offers the largest coverage area, especially, its coverage area tends to infinity when the sample size is less than 10, 12 and 19 at specific confidence level 0.90, 0.95 and 0.99, respectively.

關鍵字

coverage probability ； coverage area ； Monte Carlo simulation

並列摘要

令X_1, X_2, ..., X_n為獨立且具相同分配之常態隨機變數,其平均數μ與變異數φ均為未知。本研究的目的為同時處理有關μ與φ的100(1 - α)%信賴區域的問題。一般常見的三種以概似值(likelihood)為基礎的建構方法(LCR 法, WCR 法及SCR 法)分別是將概似比檢定,Wald 檢定及Rao 分數檢定所對應的接受區域轉換成漸進信賴區域。令X 與S^2分別表示樣本大小為n的常態隨機樣本之樣本平均數及樣本變異數, 則由(i) T_1 = n(X - μ)^2/φ∼ X_1^2, (ii) T_2 = (n-1)S^2/φ∼ X_(n-1)^2 及(iii) T_1與T_2互為獨立, 吾人可建立一個100(1 - α)%精確聯合信賴區域(JCR 法)。此外, 亦可利用Bonferroni 不等式(BCR 法)建立μ與φ的聯合信賴區間。本研究藉由蒙地卡羅模擬比較上述方法的涵蓋機率及涵蓋面積以尋求一個較為合適的方法。模擬結果顯示JCR 法在多數的樣本大小考量下其涵蓋機率都能較準確地達到所設定的名目信賴水準, 而LCR 法(BCR 法) 在小樣本至中度樣本的情況下其涵蓋機率都略低於(高於) 所設定的名目信賴水準。此外WCR 法雖具有最小的涵蓋面積但其涵蓋機率卻明顯地低於所設定的名目信賴水準。另一方面, SCR 法在這些方法中具有最大的涵蓋面積, 特別是當名目信賴水準設定在0.90, 0.95 及0.99 而其樣本數卻分別低於10, 12 及19 時會使得其涵蓋面積趨向無窮大。

並列關鍵字

涵蓋機率 ； 涵蓋面積 ； 蒙地卡羅模擬