Results concerning the generalized partially linear single-index model

The central topic of this article is the estimation of parameters of the generalized partially linear single-index model (GPLSIM). Two numerical optimization procedures are presented and an S-plus program based on these procedures is compared to a program by Wand in a simulation setting. The results from these simulations indicate that the estimates for the new procedures are as good, if not better, than Wand's. Also, this program is much more flexible than Wand's since it can handle more general models. Other simulations are also conducted. The first compares the effects of using linear interpolation versus spline interpolation in an optimization procedure. The results indicate that by using spline interpolation one gets more stable estimates at a cost of increased computational time. A second simulation was conducted to assess the performance of a method for estimating the variance of alpha. A third set of simulations is carried out to determine the best criterion for testing that one of the elements of alpha is equal to zero. The GPLSIM is applied to a water quality data set and the results indicate an interesting relationship between gastrointestinal illness and turbidity (cloudiness) of drinking water.     

Model selection and model averaging for semiparametric partially linear models with missing data

We study model selection and model averaging in semiparametric partially linear models with missing responses. An imputation method is used to estimate the linear regression coefficients and the nonparametric function. We show that the corresponding estimators of the linear regression coefficients are asymptotically normal. Then a focused information criterion and frequentist model average estimators are proposed and their theoretical properties are established. Simulation studies are performed to demonstrate the superiority of the proposed methods over the existing strategies in terms of mean squared error and coverage probability. Finally, the approach is applied to a real data case.     

A Bayesian multivariate partially linear single-index probit model for ordinal responses

Combining the multivariate probit models with the multivariate partially linear single-index models, we propose new semiparametric latent variable models for multivariate ordinal response data. Based on the reversible jump Markov chain Monte Carlo technique, we develop a fully Bayesian method with free-knot splines to analyse the proposed models. To address the problem that the ordinary Gibbs sampler usually converges slowly, we make use of the partial-collapse and parameter-expansion techniques in our algorithm. The proposed methodology are demonstrated by simulated and real data examples.     

Sensitivity analysis of partially linear models with response missing at random

This article investigates case-deletion influence analysis via Cook’s distance and local influence analysis via conformal normal curvature for partially linear models with response missing at random. Local influence approach is developed to assess the sensitivity of parameter and nonparametric estimators to various perturbations such as case-weight, response variable, explanatory variable, and parameter perturbations on the basis of semiparametric estimating equations, which are constructed using the inverse probability weighted approach, rather than likelihood function. Residual and generalized leverage are also defined. Simulation studies and a dataset taken from the AIDS Clinical Trials are used to illustrate the proposed methods.     

CBPS-based estimation for linear models with responses missing at random

In this article, based on the covariate balancing propensity score (CBPS), estimators for the regression coefficients and the population mean are obtained, when the responses of linear models are missing at random. It is proved that the proposed estimators are asymptotically normal. In simulation studies and real example, the proposed estimators show improved performance relative to usual augmented inverse probability weighted estimators.     

Inverse probability weighted estimators for single-index models with missing covariates

Abstract

In this article, we consider the inverse probability weighted estimators for a single-index model with missing covariates when the selection probabilities are known or unknown. It is shown that the estimator for the index parameter by using estimated selection probabilities has a smaller asymptotic variance than that with true selection probabilities, thus is more efficient. Therefore, the important Horvitz-Thompson property is verified for the index parameter in single index model. However, this difference disappears for the estimators of the link function. Some numerical examples and a real data application are also conducted to illustrate the performances of the estimators.     

Semiparametric estimation for weighted average derivatives with responses missing at random

When responses are missing at random, we propose a semiparametric direct estimator for the missing probability and density-weighted average derivatives of a general nonparametric multiple regression function. An estimator for the normalized version of the weighted average derivatives is constructed as well using instrumental variables regression. The proposed estimators are computationally simple and asymptotically normal, and provide a solution to the problem of estimating index coefficients of single-index models with responses missing at random. The developed theory generalizes the method of the density-weighted average derivatives estimation of Powell et al. (1989) for the non-missing data case. Monte Carlo simulation studies are conducted to study the performance of the methods.     

Variable selection for semiparametric varying coefficient partially linear model based on modal regression with missing data

Abstract

In this article, we focus on the variable selection for semiparametric varying coefficient partially linear model with response missing at random. Variable selection is proposed based on modal regression, where the non parametric functions are approximated by B-spline basis. The proposed procedure uses SCAD penalty to realize variable selection of parametric and nonparametric components simultaneously. Furthermore, we establish the consistency, the sparse property and asymptotic normality of the resulting estimators. The penalty estimation parameters value of the proposed method is calculated by EM algorithm. Simulation studies are carried out to assess the finite sample performance of the proposed variable selection procedure.     

Empirical likelihood for the response mean of generalized linear models with missing at random responses

In this article, we present a new empirical likelihood ratio for constructing the confidence interval of the response mean of generalized linear models with missing at random responses. Compared with the existing methods, the proposal can avoid the so-called “curse of dimensionality” problem when the dimension of covariates is high, and is still chi-squared distributed asymptotically, nevertheless. Simulation studies are also provided to illustrate the performance of the developed method.     

Semiparametric inference on partially linear single-index model

ABSTRACT

We consider semiparametric inference on the partially linearsingle-index model (PLSIM). The generalized likelihood ratio (GLR) test is proposed to examine whether or not a family of new semiparametric models fits adequately our given data in the PLSIM. A new GLR statistic is established to deal with the testing of the index parameter α₀ in the PLSIM. The newly proposed statistic is shown to asymptotically follow a χ²-distribution with the scale constant and the degrees of freedom being independent of the nuisance parameters or function. Some finite sample simulations and a real example are used to illustrate our proposed methodology.     

Using local linear kernel smoothers to test the lack of fit of nonlinear regression models

Herein, we propose a data-driven test that assesses the lack of fit of nonlinear regression models. The comparison of local linear kernel and parametric fits is the basis of this test, and specific boundary-corrected kernels are not needed at the boundary when local linear fitting is used. Under the parametric null model, the asymptotically optimal bandwidth can be used for bandwidth selection. This selection method leads to the data-driven test that has a limiting normal distribution under the null hypothesis and is consistent against any fixed alternative. The finite-sample property of the proposed data-driven test is illustrated, and the power of the test is compared with that of some existing tests via simulation studies. We illustrate the practicality of the proposed test by using two data sets.     

Statistical inferences for partially linear single-index models with error-prone linear covariates

We consider statistical inference for partially linear single-index models (PLSIM) when some linear covariates are not observed, but ancillary variables are available. Based on the profile least-squared estimators of the unknowns, we study the testing problems for parametric components in the proposed models. It is to see whether the generalized likelihood ratio (GLR) tests proposed by Fan et al. (2001) are applicable to testing for the parametric components. We show that under the null hypothesis the proposed GLR statistics follow asymptotically the χ²-distributions with the scale constants and the degrees of freedom being independent of the nuisance parameters or functions, which is called the Wilks phenomenon. Simulated experiments are conducted to illustrate our proposed methodology.     

Semi-empirical pseudo-likelihood for estimating equations in the presence of missing responses

Consider a semiparametric model which parameterizes only the conditional distribution of Y given X, f(y|x,β), and allows the marginal distribution of X to be completely arbitrary. Under the semiparametric model, we develop semi-empirical pseudo-likelihood inference with estimating equation in the presence of missing responses. We define semi-empirical likelihood pseudo-score estimates for both the model parameter and the parameter in the estimating equation simultaneously. Also, we develop semi-empirical pseudo-likelihood ratio inference for them, respectively. A simulation was conducted to evaluate the finite sample properties of the proposed estimators and semi-empirical pseudo-likelihood approach.     

Robust modal estimation and variable selection for single-index varying-coefficient models

In this article, we present a new efficient iteration estimation approach based on local modal regression for single-index varying-coefficient models. The resulted estimators are shown to be robust with regardless of outliers and error distributions. The asymptotic properties of the estimators are established under some regularity conditions and a practical modified EM algorithm is proposed for the new method. Moreover, to achieve sparse estimator when there exists irrelevant variables in the index parameters, a variable selection procedure based on SCAD penalty is developed to select significant parametric covariates and the well-known oracle properties are also derived. Finally, some numerical examples with various distributed errors and a real data analysis are conducted to illustrate the validity and feasibility of our proposed method.     

Robust confidence regions for the semi-parametric regression model with responses missing at random

In this paper, a regression semi-parametric model is considered where responses are assumed to be missing at random. From the empirical likelihood function defined based on the rank-based estimating equation, robust confidence intervals/regions of the true regression coefficient are derived. Monte Carlo simulation experiments show that the proposed approach provides more accurate confidence intervals/regions compared to its normal approximation counterpart under different model error structure. The approach is also compared with the least squares approach, and its superiority is shown whenever the error distribution in the simulation study is heavy tailed or contaminated. Finally, a real data example is given to illustrate our proposed method.     

On Variance-Stabilizing Multivariate Non Parametric Regression Estimation

The mean squared error (MSE)-minimizing local variable bandwidth for the univariate local linear estimator (the LL) is well-known. This bandwidth does not stabilize variance over the domain. Moreover, in regions where a regression function has zero curvature, the LL estimator is discontinuous. In this paper, we propose a variance-stabilizing (VS) local variable diagonal bandwidth matrix for the multivariate LL estimator. Theoretically, the VS bandwidth can outperform the multivariate extension of the MSE-minimizing local variable scalar bandwidth in terms of asymptotic mean integrated squared error and can avoid discontinuity created by the MSE-minimizing bandwidth. We present an algorithm for estimating the VS bandwidth and simulation studies.     

Empirical likelihood inference for partial functional linear model with missing responses

In this paper, we consider the empirical likelihood inferences of the partial functional linear model with missing responses. Two empirical log-likelihood ratios of the parameters of interest are constructed, and the corresponding maximum empirical likelihood estimators of parameters are derived. Under some regularity conditions, we show that the proposed two empirical log-likelihood ratios are asymptotic standard Chi-squared. Thus, the asymptotic results can be used to construct the confidence intervals/regions for the parameters of interest. We also establish the asymptotic distribution theory of corresponding maximum empirical likelihood estimators. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths of confidence intervals. An example of real data is also used to illustrate our proposed methods.     

A link-free sparse group variable selection method for single-index model

For regression problems with grouped covariates, we adapt the idea of sparse group lasso (SGL) [10 J. Friedman, T. Hastie, and R. Tibshirani, A note on the group lasso and a sparse group lasso, Tech. Rep., Statistics Department, Stanford University, 2010. [Google Scholar]] to the framework of the sufficient dimension reduction. Assuming that the regression falls into a single-index structure, we propose a method called the sparse group sufficient dimension reduction to conduct group and within-group variable selections simultaneously without assuming a specific link function. Simulation studies show that our method is comparable to the SGL under the regular linear model setting and outperforms SGL with higher true positive rates and substantially lower false positive rates when the regression function is nonlinear. One immediate application of our method is to the gene pathway data analysis where genes naturally fall into groups (pathways). An analysis of a glioblastoma microarray data is included for illustration of our method.