由伯氏多項式對形狀限制的回歸函數定義最大概似估計量

Maximum likelihood estimation for a shape-restricted regression model by sieve of Bernstein polynomials

潘君豪(Chun-hao Pan)

指導教授 ： 趙一峰 ； 張憶壽

國立中央大學/理學院/數學系/博士(2012年)

摘要

我們藉由伯氏多項式的次方和係數來對一個回歸函數定義最大概似估計量。如果我們已知回歸函數滿足某些形狀上的限制，例如單調性或凸性，則我們就可以透過對伯氏多項式的係數增加一樣的限制使得估計量達到相同的形狀限制。對於此類的最大概似估計量，當回歸函數連續時可建立出此估計量的收斂性；當回歸函數的導函數滿足利普希茨連續性時則可建立出此估計量的收斂速度。也是在一樣的條件下，估計量的積分也會弱收斂到回歸函數的積分。模擬分析展現出此方法在數值上的結果，除了對回歸函數的積分有良好的信賴區間的估計之外，此法亦表現得比貝氏方法及密度-回歸法更好(見Chang et al.(2007))。

關鍵字

none

並列摘要

We consider maximum likelihood estimation (MLE) of a regression function using sieves defined by Bernstein polynomials, in terms of their order and coefficients. In case, that we know the regression function satisfies certain shape-restriction like monotonicity or convexity, we can impose corresponding restriction through the coefficients of the Bernstein polynomials in the sieves so that the estimate also satisfies the desired shape-restriction. For sieve MLE of this type, we establish its consistency when the regression function is continuous and its rate of convergence when its derivative satisfies Lipschitz condition. Under the same condition, we also show that the integral of the estimate converges weakly to that of the regression function at rate of root n. Simulation studies are presented to evaluate its numerical performance. In addition to excellent confidence interval estimates of area under the regression function, sieve MLE performs better than the Bayesian method based on Bernstein polynomials and density-regression method, reported in Chang et al. (2007).

並列關鍵字

Bernstein polynomials ； Area under the curve ； rate of convergence ； shape -restricted regression ； sieve maximum likelihood estimate. ； empirical process

參考文獻

[1] Altshuler, B (1981). Modeling of dose-response relationships. Scandinavian Journal of statisticsEnviron. Healt Perspect. 42 23-27.

[2] Birke, M. and Dette, H. (2007). Estimating a convex function in nonparametric regression. Scandinavian Journal of statistics. 34 384-404.

[3] Chang, I. S., Chien, L. C., Hsiung, C. A., Wen, C. C. and Wu, Y. J. (2007). Shape restricted regression with random Bernstein polynomials. Complex Datasets and Inverse Problems: Tomography, Networks and Beyond. 54 187–202.

[4] Chang, I. S., Hsiung, C. A., Wu, Y. J. and Yang, C. C. (2005). Bayesian survival analysis using Bernstein polynomials. Scand. J. Statist. 32 447–466.

[5] Chak, P. M., Madras, N. and Smith, B. (2005). Semi-nonparametric estimation with Bernstein polynomials. Economics Letters. 89 153-156.