臺灣博碩士論文加值系統

在無母數迴歸方法中，樣條平滑(smoothing spline)法常被使用在估計右設限存活資料的風險函數，而此方法的關鍵在於平滑參數的選取。在過去的文獻中，有許多學者提出選取平滑參數的準則。例如:交叉驗證法(cross validation)，廣義交叉驗證法(generalized cross validation)。然而，這兩種準則並不是以降低均方差為前提下所發展的。因此，我們提出一準則，透過Ye (1998)的資料擾動法(date perturbation)和廣義自由度(generalized degrees of freedom)的概念，估計均方差的Stein不偏風險估計量，來選取平滑參數。藉由模擬方式來說明所提方法的有效性，並且應用所提方法於兩組實例資料。

Smoothing spline is a popular technique to estimate the hazard function for right-censored lifetime data. When using smoothing spline, selecting the smoothing parameter is an important issue. The cross-validation and generalized cross-validation methods are frequently used to estimate the smoothing parameters. However, these two criteria are not developed for minimizing of the mean square error. In this work, we propose a method based on the Stein’s unbiased risk estimate of the mean square error and the concept of generalized degrees of freedom which are obtained via a data perturbation technique. The performance of the proposed method is justified by a simulation study. Finally the method is applied to two real data sets.

1.研究目的
2.文獻回顧
2.1 樣條平滑法
2.2 選取平滑參數
2.2.1 交叉驗證法
2.2.2廣義交叉驗證法
3.研究方法
4.模擬研究
4.1模擬設定
4.2模擬流程
4.3模擬結果
5.實例分析
5.1多發性骨髓瘤研究資料
5.2德國乳腺癌研究資料
6.結論與建議
參考文獻

Craven, P., Wahba, G., 1979. Smoothing noisy data with spline funciton: estimating the correct degree of smoothing by the method of generalized cross-validation. Numer. Math. 31, 377-403.
Gonzàlez-Manteiga, W., Cao, R., Marron, J.S., 1996. Bootstrap Selection of the Smoothing Parameter in Nonparametric Hazard Rate Estimation. Journal of the American Statistical Association. 91, 1130-1140.
Good, I.J., Gaskins, R.A., 1971. Nonparametric roughness penalties for probability densities. Biometrics. 58, 255-277.
Huber, P.T., 1981. Robust Statistics. New York. Wiley.
Krall, J.M., Uthoff, V.A., Harley, J.B., 1975. A step-up procedure for selecting variables associated with survival. Biometrics. 31(1), 49-57.
Li, K.C., 1989. From Stein's Unbiased Risk Estimates to the Method of Generalized Cross Validation. Neural Computation. 2, 281-294.
Rice, J., Silverman, B.W., 1991. Estiamting the Mean and Covariance Structure nonparametrically When the Data are Curves. Journal of the royal Statistical society Series B. 53(1), 233-243.
Rosenberg, P.S., 1995. Hazard function estimation using B-splines. Biometrics. 51, 874-887.
Sauerbrei, W., Royston, P., 1999. Building multivariable prognositc and diagnostic models: transformation of eh predictors using fractional polynomials. Journal of the Royal Statistical Society Series A. 162, 1, 71-94.
Stein, C., 1981. Estimation of the Mean of a Multivariate Normal Distribution. The Annals of Statistics. 9, 1135-1151.
Tanner, M.A., Wong, W.H., 1983. The Estimation of the Hazard Function form Randomly censored Data by the Kernel Method. The Annals of Statistics. 1, 989-993.
Wand, M.P., Jones, M.C., 1995. Kernel Smothing. Chapman & Hall, New York.
Ye, J., 1998. On Measuring and Correcting the Effects of Data Mining and Model Selection. Journal of the American Statistical Association. 93, 120-131.
Ye, J., Shen, X., Huang, H.C., 2004. Comment on “The estimation of prediction error: covariance penalties and cross validation” by B. Efron. Journal of the American Statistical Association. 99, 634-637