### Abstract

In this thesis, we firstly develop a model-adaptive checking method for partially parametric single-index models, which combines the advantages of both dimension reduction technique and global smoothing tests. Besides, we propose a dimension reduction-based model adaptive test of heteroscedasticity checks for nonparametric and semi-parametric regression models. Finally, to extend our testing approaches to nonparametric regressions with some restrictions, we consider significance testing under a nonparametric framework. In Chapter 2, “Model Checking for Partially Parametric Single-index Models: A Model-adaptive Approach", we consider the model checking problems for more general parametric models which include generalized linear models and generalized nonlinear models. We develop a model-adaptive dimension reduction test procedure by extending an existing directional test. Compared with traditional smoothing model checking methodologies, the procedure of this test not only avoids the curse of dimensionality but also is an omnibus test. The resulting test is omnibus adapting the null and alternative models to fully utilize the dimension-reduction structure under the null hypothesis and can detect fully nonparametric global alternatives, and local alternatives distinct from the null model at a convergence rate as close to square root of the sample size as possible. Finally, both Monte Carlo simulation studies and real data analysis are conducted to compare with existing tests and illustrate the finite sample performance of the new test. In Chapter 3,Heteroscedasticity Checks for Nonparametric and Semi-parametric Regression Model: A Dimension Reduction Approach", we consider heteroscedasticity checks for nonparametric and semi-parametric regression models. Existing local smoothing tests suffer severely from the curse of dimensionality even when the number of covariates is moderate because of use of nonparametric estimation. In this chapter, we propose a dimension reduction-based model adaptive test that behaves like a local smoothing test as if the number of covariates is equal to the number of their linear combinations in the mean regression function, in particular, equal to 1 when the mean function contains a single index. The test statistic is asymptotically normal under the null hypothesis such that critical values are easily determined. The finite sample performances of the test are examined by simulations and a real data analysis. In Chapter 4，Dimension Reduction-based Significance Testing in Nonparametric Regression", as nonparametric techniques need much less restrictive conditions than those required for parametric approaches, we consider to check nonparametric regressions with some restrictions under sufficient dimension reduction structure. A dimension-reduction-based model-adaptive test is proposed for significance of a subset of covariates in the context of a nonparametric regression model. Unlike existing local smoothing significance tests, the new test behaves like a local smoothing test as if the number of covariates is just that under the null hypothesis and it can detect local alternative hypotheses distinct from the null hypothesis at the rate that is only related to the number of covariates under the null hypothesis. Thus, the curse of dimensionality is largely alleviated when nonparametric estimation is inevitably required. In the cases where there are many insignificant covariates, the improvement of the new test is very significant over existing local smoothing tests on the significance level maintenance and power enhancement. Simulation studies and a real data analysis are conducted to examine the finite sample performance of the proposed test. Finally, we conclude the main results and discuss future research directions in Chapter 5. Keywords: Model checking; Partially parametric single-index models; Central mean subspace; Central subspace; Partial central subspace; Dimension reduction; Ridge-type eigenvalue ratio estimate; Model-adaption; Heteroscedasticity checks; Significance testing.Date of Award | 26 Aug 2015 |
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Original language | English |

Supervisor | Lixing ZHU (Supervisor) |

### User-Defined Keywords

- Computer adaptive testing
- Mathematical models
- Regression analysis