臺灣博碩士論文加值系統

在本文中，我們首先針對二維常態模型下，相關係數的單邊及雙邊檢定問題進行探討。這並非一個新的統計問題，事實上，在傳統數理統計已有好的檢定方法存在，例如：Lehmann (1994) 的書上所推薦的UMPU和UMPI檢定，而這些檢定的檢定統計量是以樣本相關係數作為出發。然而，對於單邊對立假設 ，其中 ，該檢定是不容易操作的。因為樣本相關係數的密度函數其型式非常複雜，因此，其檢定的臨界點是必須透過電腦程式計算的輔助才得以計算出。而在實際應用層面上，當 時，顯著地建立此種對立假設 對於決策的擬定是很有實用上的價值。但是，傳統的UMPI檢定方法無法讓我們立即地得到檢定的結論，於是其存在著被改進的空間，這也是本研究主要的動機。

因此，本研究主要的目標是對於任意給定的 ，尤其是較大的 ，建構出較佳且容易操作的水準α之檢定。首先，我們利用Berger and Sun (2008) 所提出相關係數的信賴區間，透過信賴區間和檢定一體兩面的關係，找出其信賴區間所對應的檢定。其次，在加上兩個方差參數為相同的假設下，我們推薦了一個新的UMPU檢定。

透過數值計算和模擬研究的結果，我們首先觀察到Berger and Sun (2008) 所提出的信賴區間其對應的檢定和傳統的UMPI檢定方法，在檢定力的表現上幾乎一致。其次，我們所推薦的檢定在檢定力的表現上，皆比傳統的UMPI檢定及Berger and Sun (2008) 所提的信賴區間所對應的檢定方法來得要好。在許多實用層面重要的情況有非常明顯地改善，甚至超過100%的改善率。然而，在兩個方差參數不見得相同的情況，我們也提出調整我們所提檢定的方式，進而有我們推薦的修正檢定。模擬結果發現，在樣本數不是太小時，此修正檢定的水準雖會些微高於所宣稱值，但在檢定力上是有實質增加，是優於傳統檢定的；實務應用上是值得推薦的。

另外，我們在二維偏斜常態模型架構下，對於此三種檢定方法的穩健性進行探討。透過大量的模擬研究，我們發現此三種檢定的檢定力函數會隨著所宣稱的水準而有激進或保守的偏離狀況，而偏離程度會依偏斜係數越大而越嚴重、隨著樣本數越大而越嚴重。因此，在偏斜常態分配架構下，此三種方法皆不夠穩健。

We consider the testing problem, one-sided or two-sided, about the correlation coefficient ρ of a bivariate normal distribution. This is not a new inferential problem and good tests exist in the literature, for example, Lehmann (1994) gives the corresponding UMPU and UMPI tests which are based on the sample correlation coefficient R. However, for the alternative H1:ρ>ρ0 with ρ0≠0, the tests above are not easy to implement since their critical points need to be calculated numerically, due to the complicated density function of R. Significantly establish H1:ρ>ρ0 with ρ0 >>0 may be valuable for decision making in some applications, yet one can not get the conclusion of the classical UMPI immediately ─ this motivates our study here.

Hence, in this work, we aim at constructing better level α tests which are easier to implement for any givenρ0, especially for large |ρ0 |. First, we make use of a new confidence interval forρprovided by Berger and Sun (2008) to construct new tests through the duality relationship between confidence interval and tests. Next, we propose new UMPU tests by adding the equal variance assumption.

By the results of simulations and numerical calculations, we notice first that the power functions of the tests by inverting confidence intervals provided by Berger and Sun (2008) are almost identical to the classical UMPI tests; secondly, when equal variances assumption holds, our test is theoretically proved to have larger power function than the classical UMPI test and numerical results indicate that the improvement rate can be more than 100%, quite substantial, in some practically important cases. When variances are not equal, we provide a modified version of our test. Numerical results indicates that, for moderate to large sample sizes, our modified
test may have a slightly larger level than the nominal level, yet, in such cases, it has larger power functions than the UMPI and the improvement can be substantial -- acceptable to many practioners.

Moreover, we study the robustness of these tests under a bivariate skew-normal distribution. By the results of simulations, we notice that the power functions of these tests all fail to retain the claimed level. The discrepancies are large as the skew parameters become larger or as the sample sizes become larger. Hence, these tests are not robust against bivariate skew-normality.

1 Introduction
2 Tests for correlation coefficient of bivariate normal
2.1 One-sided tests
2.2 Two-sided tests
2.3 What if σx≠σy?
3 Robustness of these tests against skew-normality

Azzalini, A. (2005). The skew-normal distribution and related multivariate families. Scandinavian Journal of Statistics, 159-188.

Berger, J. O. and Sun, D. (2008). Objective priors for the bivariate normal model. Annals of Statistics. 36, 963-982.

Lehmann, E. L. (1994). Testing Statistical Hypotheses, 2nd edition. Chapman and Hall.

Liao, C. C. (2009). Robustness of confidence intervals for a normal mean and some interval estimators, powerful unbiased tests under skew-normal model. Master thesis, Department of Applied Mathematics,National Dong Hwa University.