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 比例型態隨機變數的估計與推論常應用於不同的統計問題上,例如: 線性模式之反預測, 邊際藥劑量, 相關效力等問題;在衰退測試實驗中，線性衰退路徑的隨機效果亦構成此種型態之隨機變數.伯恩斯坦式是建立於描述切割工具壽命而產生的,已被工程界廣泛運用於描述元件壽命.我們以傳統的估計方法, %calibration華德統計量和最大概似比檢定對比例型態問題推論與估計,並和伯恩斯坦模式相比較.當衰退模式中斜率項為正之假設不成立時,Hinkley (1969) 所整理二項常態比例之分配,提供了一個精確的分配形態.在伯恩斯坦模式中,線性衰退路徑之截距與斜率推廣至二項分配之形態,並進一步討論其他分配型態的截距與斜率之情況.
 The inference of ratio-typed random variables occurred often in different application situations,for example calibration, critical dosage,relative potency, etc. Estimation of failure time distribution in linear degradation model with random intercept and slope also involves ratio of two random variables. Bernstein model has been established in degradation analysis in estimating a time-to-failure distribution. In this work, we study the approaches of Wald and maximum likelihood, compared with Bernstein distribution, in estimation and making inference of ratio-typed quantity of interest. Profile likelihood is used to construct confidence interval in likelihood approach.The distribution of the ratio of bivariate normally distributed random variables is discussed by Hinkley (1969),it provides an exact and correct distribution of Bernstein model when the assumption of positive slope is not hold.
 1 Introduction 31.1 Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Overview . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 42 Dose Response Model 62.1 MLE of Dose Response Model Parameters . . . . . . . . . . . . . . . . . 62.2 Reparameterize Critical Lethal Dose in Likelihood Function . . . . . . .72.3 Adequacy of Model Selection and Comparison . . . . . . . . . . . . . . 102.4 Calibration Estimates of Critical Lethal Dose . . . . . . . . . . .. . 122.4.1 Calibration Con dence Interval . . . . . . . . . . . . . . . . . . . 133 Bernstein Model 173.1 Fatigue-Crack-Growth Data . . . . . . . . . . . . . . . . . . . . . . 173.2 Parameterization in Ahmad and Sheikh (1983) . . . . . . . . . . . . . .183.3 Derivation of Bernstein Distribution . . . . . . . . . . . . . . . . . 213.4 Reparameterization . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4.1 Properties and Special Cases for Bernstein Model . . . . . . . . . . 243.5 Simpli ed Bernstein Model . . . . . . . . . . . . . . . . . . . . . . 253.5.1 Simpli ed Bernstein Q-Q plot . . . . . . . . . . . . . . . . . . . . 253.5.2 Maximum Likelihood Estimates of Parameters . . . . . . . . . . . . . 293.5.3 Likelihood-Based Con dence Region and Con dence Interval . . . . . . 303.6 Three-parameter Bernstein Model . . . . . . . . . . . . . . . . . . . .334 Generalized and Extended Bernstein Model 364.1 The Case of Correlated Intercept and Slope . . . . . . . . . . . . . . 364.2 Application of Bernstein Distribution . . . . . . . . . . . . . . . . . 394.2.1 Dose Response Model . . . . . . . . . . . . . . . . . . . . . . . . . 394.2.2 Fatigue-Crack-Growth Data . . . . . . . . . . . . . . . . . . . . . . 404.3 Quantile Function of Bernstein Distribution . . . . . . . . . . . . . . 424.4 Exact Distribution of X=Y . . . . . . . . . . . . . . . . . . . . . . . 444.4.1 Approximate Bernstein distribution . . . . . . . . . . . . . . . . . .444.4.2 Hinkley's Expression . . . . . . . . . . . . . . . . . . . . . . . . .454.5 Numeric Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . 484.6 Extended Bernstein Distribution . . . . . . . . . . . . . . . . . . . . 514.6.1 Fixed Intercept and Random Slope . . . . . . . . . . . . . . . . . . .534.6.2 Random Intercept and Fixed Slope . . . . . . . . . . . . . . . . . . .545 Concluding Remarks 56References 58R Functions 60
 Ahmad, M., and Sheikh, A. K. (1984), ìbernstein reliability model: Derivation and estimationof parameters,î Reliability Engineering, 8, 131ñ148.Bliss, C. I. (1935), ìThe calculation of the dosage-mortality curve,î Annals of AppliedBiology, 22, 134ñ167.Bogdanoff, J. L., and Kozin, F. (1985), Probabilistic Models of Cumulative Damage, NewYork: John Wiley & Sons.Chen, S. F. (1999), Bernstein Model in Accelerated Degradation Testing and Analysis, Departmentof statistics, National Cheng Kung University, Tainan, Taiwan, unpublishedmaster thesis.Dobson, A. J. (1990), An Introduction to Generalized Linear Models, London: Chapman &Hall.Fieller, E. C. (1932), ìThe distribution of the index in a normal bivariate population,îBiometrika, 24, 428ñ440.ó (1954), ìSome problems in interval estimation,î Journal of the Royal Statistical Society,12, 175ñ185.Fisch, R. D., and Strehlau, G. A. (1993), ìA simpli ed approach to calibration con dencesets,î The American Statistician, 47, 168ñ171.Fisher, R. A. (1970), Statistical Methods for ResearchWorkers, Edinburgh: Oliver and Boyd,14th edition.Gertsbakh, I. B., and Kordonsky, K. B. (1969), Models of Failure, New York: Springer-Verlag.Graybill, F. A. (1976), Theory and Application of the Linear Model, Belmont, CA:Wadsworth.Hinkley, D. V. (1969), ìOn the ratio of two correlated normal random variables,îBiometrika, 56, 635ñ639.Hwang, J. T. G. (1995), ìFieller's problems and resampling techniques,î Statistica Sinica,5, 161ñ171.Krutchkoff, R. G. (1967), ìClassical and universe methods of calibration,î Technometrics,11, 425.Lu, C. J., and Meeker, W. Q. (1993), ìUsing degradation measures to estimate a time-tofailuredistribution,î Technometrics, 35, 161ñ174.Meeker, W. Q., and Escobar, L. A. (1995), ìTeaching about approximate con dence regionsbased on maximum likelihood estimation,î The American Statistician, 49, 48ñ53.ó (1998), Statistical Methods for Reliability Data, New York: John Wiley & Sons.Seber, G. A. F. (1977), Linear Regression Analysis, New York: John Wiley & Sons.Wang, C. C. (1994), The study of requiring sample in estimating population percentile in reliabilityanalysis, Department of statistics, National Cheng Kung University, Tainan,Taiwan, unpublished master thesis.Williams, E. J. (1969), ìA note on regression methods in calibration,î Technometrics, 11,189ñ192.
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