Estimation of smooth regression functions in monotone response models
- 作者：Jayanta Kumar Pal ; Moulinath Banerjee
- 关键词：Monotone response models ; Smoothing spline ; Confidence interval
- 刊名：Journal of Statistical Planning and Inference
- 出版年：2008
- 期刊代码：36_03783758
- 类别：mt
- 出版时间：1 October 2008
- 卷：138
- 期：10
- 页码：3125-3143
- 文件大小：321 K
摘要
We consider the estimation of smooth regression functions in a class of conditionally parametric co-variate-response models. Independent and identically distributed observations are available from the distribution of (Z,X), where Z is a real-valued co-variate with some unknown distribution, and the response X conditional on Z is distributed according to the density p(·,ψ(Z)), where p(·,θ) is a one-parameter exponential family. The function ψ is a smooth monotone function. Under this formulation, the regression function E(X|Z) is monotone in the co-variate Z (and can be expressed as a one–one function of ψ); hence the term “monotone response model”. Using a penalized least squares approach that incorporates both monotonicity and smoothness, we develop a scheme for producing smooth monotone estimates of the regression function and also the function ψ across this entire class of models. Point-wise asymptotic normality of this estimator is established, with the rate of convergence depending on the smoothing parameter. This enables construction of Wald-type (point-wise) as well as pivotal confidence sets for ψ and also the regression function. The methodology is extended to the general heteroscedastic model, and its asymptotic properties are discussed.