Statistical Inference For Semiparametric Models Under Shape Constraints

In the ?elds of economics, biology and sociology, the prior information between the response variable and the covariates is often available. This information will make statistical inference more e?cient and reasonable. For example, in children’s growth curve study, height increases with age, so the growth curve is of monotone nondecreasing. In survival analysis hazard rate function has various shapes, such as increasing, decreasing, constant, convex, etc. A decreasing hazard rate may arise due to natural aging or wearing, while most population mortality data is subject to convex constraints. In view of this, when we make statistical inference, these constraints should be placed in the statistical model. When statistical inference problems have constraints, unconstrained statistical methods, which often produce results that does not satisfy the constraints, may not be suitable.So it requires a new approach to solve these problems.In this dissertation, we mainly discuss statistical inference for the semiparametric partially linear model under shape constrains. The contents include the classical estimation, Bayesian estimation and statistical tests for shape constrains.The contents are organized as following.Introduction is in Chapter 1, we introduce the research contents, research status and existing methods.In Chapter 2, a Bernstein-polynomial-based likelihood method is proposed for the partially linear model under monotonicity constraint. Monotone Bernstein polynomials are employed to approximate the monotone nonparametric function in the model. The estimator of the regression parameter is shown to be asymptotically normal and e?cient, and the rate of convergence of the estimator of the nonparametric component is established, which could be the optimal under the smooth assumptions. A simulation study and a real data analysis are conducted to evaluate the ?nite sample performance of the proposed method.In Chapter 3, we propose Bernstein polynomial estimation for the partially linear model when the nonparametric component is subject to convex(or concave)constraint. We employ a nested sequence of Bernstein polynomials to approximate the convex(or concave) nonparametric function. We show that the estimator of the parametric part is asymptotically normal. The rate of convergence of the nonparametric function estimator is established under very mild conditions. The small sample properties of our estimation are provided via simulation study and compared with regression splines method. A real data analysis is conducted to illustrate the application of the proposed method.In Chapter 4, we develop M-estimation for the partially linear model in which the nonparametric component is subject to various shape constraints. Bernstein polynomials are used to approximate the unknown nonparametric function, and shape constraints are imposed on the coe?cients. The general loss function is used for robustness of the estimation. Asymptotic normality of regression parameters and the optimal rate of convergence of the shape-restricted nonparametric function estimator are established under very mild conditions. Some simulation studies and a real data analysis are conducted to evaluate the ?nite sample performance of the proposed method.In Chapter 5, we develop a Bayesian estimation method to nonparametric models and nonparametric mixed-e?ects models under shape-constrains. The approach uses the characterizations of shape-constrained Bernstein polynomials. We combine Gibbs sampler and Metropolis-Hastings sampler for model ?tting, using a truncated normal distribution as the prior for the coe?cients of Bernstein polynomial to ensure the desired shape constraints. The small sample properties of the Bayesian shape-constrained estimators across a range of functions are provided via simulation study and compared with other methods. A real data analysis is conducted to illustrate the application of the proposed method.In Chapter 6, we tend to explore the tests of signi?cance for parametric components and tests of monotonicity and convexity in the partially linear model, and a test statistics is developed based on the derivative of the unconstrained estimator. The Bootstrap method is used to approximate the distribution of the test statistics. Our proposed test is compared with the other test statistics though the simulations.