The Study For Several Statistical Inference Issues In Single-Index Model

The single-index model, which is a potentially tool for dealing with the multivariate non-parametric regression, is an important semiparametric model. By reducing the dimensionality from multivariate predictors to an univariate index parameters, the single-index model avoids the so-called "curse of dimensionality" in multivariate nonparametric regression and captures the important features in high-dimensional data. The statistical inference is a core of statistical research. The statistical diagnostics have become an important part which one must consider in any serious statistical analysis. In this paper, we employ some approaches, which are different with the existing methods in some literatures, to study some statistical inference issues and statistical diagnostics for single-index models. Firstly, we consider studying the statistical diagnostics in single-index models. Due to complexity of single-index models, we only investigate the local influence against some minor perturbations in model for penalized Gaussian likelihood estimations of parameters in partially linear single-index model. Some diagnostic statistics are proposed and the substantial affect resulting from all possible perturbation schemes imposed on the model is investigated for parameter estimations. A real illustrative example is explored. Secondly, we study the M-type estimators of single-index model from three view-points. One is that we use B-spline basis to approximate the unknown regression function and develop the approach obtaining the M-estimators of parameters in single-index models. Under some mild regularity conditions, the large sample properties of the M-estimator are investigated. Another is that we consider the M-type estimators based on B-spline approximation for partially linear single-index models. We develop the procedure to obtain the M-estimators and study the large sample properties of these estimators under some mild condition. The other is that we use the local linear methods and two-step estimates technique to obtain the M-estimators of parameters in partially linear single-index models. We propose an estimates procedure and prove the asymptotic properties under some conditions. Third, we consider the construction of confidence region of unknown parameter in single-index models. Based on Owen's empirical likelihood and B-spline approximation method, the estimated empirical log-likelihood ratio is obtained. Under some mild regularity conditions, we shown that the asymptotic distribution of estimated empirical log-likelihood ratio is a standard X~2-distribution. According to the result, the confidence region of unknown parameters in single-index models is constructed. Finally, we consider the test of correlation and heterogeneity for hierarchical nonlinear mixed-effects models. According to Laplace approximation expansion of integrated log-quasilikelihood, a class of score test statistics is proposed. Under some very mild conditions, we prove the asymptotic distribution of the test statistics is a standard X~2-distribution. Through a Monte Carlo simulation study, the finite sample performance is examined. A real illustrative example is explored.