本文主要是探討應用經驗分布函數對貝他-二項式模式做適合度檢定. 我們提出兩種以經驗分布函數為基礎之統計量的檢定方法, 然後透過蒙地卡羅模擬研究去檢定統計量在對立假設下之不同分布的經驗檢定力. 與 Garren et al. (2001) 的方法相比較, 我們的方法在大多數的情況下是具有較大檢定力的. 最後, 我們應用提出的檢定方法在環境毒性研究的實際資料.

Empirical-distribution-function (EDF) goodness-of-fit tests are considered for the beta-binomial model. The testing procedures based on EDF statistics are given. A Monte Carlo study is conducted to investigate the accuracy and power of the tests against various alternative distributions. Our method is found to produce considerably greater power than that of Garren et al. (2001) in most cases. The tests are applied to data sets of the environmental toxicity studies.

1 前言 1
2 Beta-Binomial 分布的適合度檢定 4
2.1 參數估計方法 . . . . . . . . . . . . . . . . . . . . 6
2.1.1 動差法 . . . . . . . . . . . . . . . . . . . 6
2.1.2 最大概似估計法 . . . . . . . . . . . . . . . 7
2.2 模擬研究 . . . . . . . . . . . . . . . . . . . . . . 11
3 Beta-Binomial 模式的適合度檢定 14
3.1 參數估計方法 . . . . . . . . . . . . . . . . . . . . 17
3.2 模擬研究 . . . . . . . . . . . . . . . . . . . . . . 19
3.3 實例分析 . . . . . . . . . . . . . . . . . . . . . . 23
參考文獻 25
附錄一 27
附錄二 30

Baringhaus, L. and Henze, N. (1992). A goodness of fit test for the Poisson distribution based on the empirical generating function. Statistics and Probability Letters 13, 269–274.

Brooks, S. P., Morgan, B. J., Bidout, M. S. and Pack, S. E. (1997). Finite mixture models for proportions. Biometrics 53, 1097–1115.

D’Agostino, R. B. and Stephens, M. A. (1986). Goodness-of-fit Techniques. Marcel Dekker, New York.

Garren, S. T., Smith, R. L. and Piegorsch, W. W. (2000). On a likelihoodbased goodness-of-fit test of the beta-binomial model. Biometrics 56, 947–949.

Garren, S. T., Smith, R. L. and Walter, W. (2001). Bootstrap goodness-of-fit test for the beta-binomial model. Journal of Applied Statistics 28, 561–571.

Griffiths, D. A. (1973). Maximum likelihood estimation for the beta-binomial distribution and an application to the household distribution of the total number of cases of a disease. Biometrics 29, 637–648.

Henze, N. (1996). Empirical-distribution-function goodness-of-fit tests for discrete models. The Canadian Journal of Statistics 24, 81–93.

Kleinman, J. C. (1973). Proportions with extraneous variance: Single and independent samples. Journal of the American Statistical Association 68, 46–54.

Kemp, C. D. and Kemp, A. W. (1956). The analysis of point quadrat data. Australian Journal of Botany 4, 167–174.

Lockhart, A.-M., Piegorsch, W. W. and Bishop, J. B. (1992). Assessing overdispersion and dose response in the male dominant lethal assay. Mutation Research 272, 35–58.

Mantel, N. and Paul, S. R. (1987). Goodness-of-fit issues in toxicological experiments involving litters of varying size. In: I. B. MacNeill and G. J. Umphrey (Eds), Biostatistics, pp. 169–176. New York, Reidel.

Skellam J. G. (1948). A probability distribution derived from the binomial distribution by regarding the probability of success as variable between the sets of trials. Journal of the Royal Statistical Society B 10, 257–261.

Smith, D. M. (1983). [Algorithm AS 189] Maximum likelihood estimation of the parameters of the beta binomial distribution. Applied Statistics 32, 196–204.

Williams, D. A. (1975). The analysis of binary responses from toxicological experiments involving reproduction and teratogenicity. Biometrics 31, 949–952.

Yamamoto, E. and Yanagimoto, T. (1992). Moment estimators for the beta-binomial distribution. Journal of Applied Statistics 19, 273–283.