Yingcun Xia

An adaptive estimation of dimension reduction space

Summary. Searching for an effective dimension reduction space is an important problem in regression, especially for high dimensional data. We propose an adaptive approach based on semiparametric models, which we call the (conditional) minimum average variance estimation (MAVE) method, within quite a general setting. The MAVE method has the following advantages. Most existing methods must undersmooth the nonparametric link function estimator to achieve a faster rate of consistency for the estimator of the parameters (than for that of the nonparametric function). In contrast, a faster consistency rate can be achieved by the MAVE method even without undersmoothing the nonparametric link function estimator. The MAVE method is applicable to a wide range of models, with fewer restrictions on the distribution of the covariates, to the extent that even time series can be included. Because of the faster rate of consistency for the parameter estimators, it is possible for us to estimate the dimension of the space consistently. The relationship of the MAVE method with other methods is also investigated. In particular, a simple outer product gradient estimator is proposed as an initial estimator. In addition to theoretical results, we demonstrate the efficacy of the MAVE method for high dimensional data sets through simulation. Two real data sets are analysed by using the MAVE approach.

Measles Metapopulation Dynamics: A Gravity Model for Epidemiological Coupling and Dynamics

Infectious diseases provide a particularly clear illustration of the spatiotemporal underpinnings of consumer‐resource dynamics. The paradigm is provided by extremely contagious, acute, immunizing childhood infections. Partially synchronized, unstable oscillations are punctuated by local extinctions. This, in turn, can result in spatial differentiation in the timing of epidemics and, depending on the nature of spatial contagion, may result in traveling waves. Measles epidemics are one of a few systems documented well enough to reveal all of these properties and how they are affected by spatiotemporal variations in population structure and demography. On the basis of a gravity coupling model and a time series susceptible‐infected‐recovered (TSIR) model for local dynamics, we propose a metapopulation model for regional measles dynamics. The model can capture all the major spatiotemporal properties in prevaccination epidemics of measles in England and Wales.

Shrinkage Estimation of the Varying Coefficient Model

The varying coefficient model is a useful extension of the linear regression model. Nevertheless, how to conduct variable selection for the varying coefficient model in a computationally efficient manner is poorly understood. To solve the problem, we propose here a novel method, which combines the ideas of the local polynomial smoothing and the Least Absolute Shrinkage and Selection Operator (LASSO). The new method can do nonparametric estimation and variable selection simultaneously. With a local constant estimator and the adaptive LASSO penalty, the new method can identify the true model consistently, and that the resulting estimator can be as efficient as the oracle estimator. Numerical studies clearly confirm our theories. Extension to other shrinkage methods (e.g., the SCAD, i.e., the Smoothly Clipped Absolute Deviation.) and other smoothing methods is straightforward.

On Single-Index Coefficient Regression Models

Abstract In this article we investigate a class of single-index coefficient regression models under dependence. This includes many existing models, such as the smooth transition threshold autoregressive (STAR) model of Chan and Tong, the functional-coefficient autoregressive (FAR) model of Chen and Tsay, and the single-index model of Ichimura. Compared to the varying-coefficient model of Hastie and Tibshirani, our model can avoid the curse of dimensionality in multivariate nonparametric estimations. Another advantage of this model is that a threshold variable is chosen automatically. An estimation method is proposed, and the corresponding estimators are shown to be consistent and asymptotically normal. Some simulations and applications are also reported.

A constructive approach to the estimation of dimension reduction directions

In this paper we propose two new methods to estimate the dimension-reduction directions of the central subspace (CS) by constructing a regression model such that the directions are all captured in the regression mean. Compared with the inverse regression estimation methods [e.g., J. Amer. Statist. Assoc. 86 (1991) 328-332, J. Amer. Statist. Assoc. 86 (1991) 316-342, J. Amer. Statist. Assoc. 87 (1992) 1025-1039], the new methods require no strong assumptions on the design of covariates or the functional relation between regressors and the response variable, and have better performance than the inverse regression estimation methods for finite samples. Compared with the direct regression estimation methods [e.g., J. Amer. Statist. Assoc. 84 (1989) 986-995, Ann. Statist. 29 (2001) 1537-1566, J. R. Stat. Soc. Ser. B Stat. Methodol. 64 (2002) 363-410], which can only estimate the directions of CS in the regression mean, the new methods can detect the directions of CS exhaustively. Consistency of the estimators and the convergence of corresponding algorithms are proved.

Bias-corrected confidence bands in nonparametric regression

Summary. Bias-corrected confidence bands for general nonparametric regression models are considered. We use local polynomial fitting to construct the confidence bands and combine the cross-validation method and the plug-in method to select the bandwidths. Related asymptotic results are obtained. Our simulations show that confidence bands constructed by local polynomial fitting have much better coverage than those constructed by using the Nadaraya‐Watson estimator. The results are also applicable to nonparametric autoregressive time series models.

ASYMPTOTIC DISTRIBUTIONS FOR TWO ESTIMATORS OF THE SINGLE-INDEX MODEL

The single-index model is one of the most popular semiparametric models in applied quantitative sciences. Two new estimation methods have been proposed recently by Hristache, Juditski, and Spokoiny ( 2001 , Annals of Statistics 29, 595–623) and Xia, Tong, Li, and Zhu ( 2002 , Journal of the Royal Statistical Society, Series B 64, 363–410), respectively. However, their asymptotic distributions have not been investigated yet. In this paper, alternative versions for the methods are investigated. Asymptotic distributions of the estimators are derived. Efficiency comparisons between the estimation methods are made. The author is most grateful to Professor O. Linton and Professor W. Hardle for helpful discussions. Valuable comments from two anonymous reviewers have improved the presentation of the paper substantially. The research has been partially supported by NUS research grant R-155-000-048-112, National University of Singapore, Singapore, and the Alexander von Humboldt Foundation, Germany.

Sliced Regression for Dimension Reduction

A new dimension-reduction method involving slicing the region of the response and applying local kernel regression to each slice is proposed. Compared with the traditional inverse regression methods [e.g., sliced inverse regression (SIR)], the new method is free of the linearity condition and has much better estimation accuracy. Compared with the direct estimation methods (e.g., MAVE), the new method is much more robust against extreme values and can capture the entire central subspace (CS) exhaustively. To determine the CS dimension, a consistent cross-validation criterion is developed. Extensive numerical studies, including a real example, confirm our theoretical findings.

Uniform Bahadur representation for local polynomial estimates of M-regression and its application to the additive model

We use local polynomial fitting to estimate the nonparametric M-regression function for strongly mixing stationary processes {(Y-i, (X) under bar (i))}. We establish a strong uniform consistency rate for the Bahadur representation of estimators of the regression function and its derivatives. These results are fundamental for statistical inference and for applications that involve plugging such estimators into other functionals where some control over higher order terms is required. We apply our results to the estimation of an additive M-regression model.

A SINGLE-INDEX QUANTILE REGRESSION MODEL AND ITS ESTIMATION

Models with single-index structures are among the many existing popular semiparametric approaches for either the conditional mean or the conditional variance. This paper focuses on a single-index model for the conditional quantile. We propose an adaptive estimation procedure and an iterative algorithm which, under mild regularity conditions, is proved to converge with probability 1. The resulted estimator of the single-index parametric vector is root- n consistent, asymptotically normal, and based on simulation study, is more efficient than the average derivative method in Chaudhuri, Doksum, and Samarov (1997, Annals of Statistics 19, 760–777). The estimator of the link function converges at the usual rate for nonparametric estimation of a univariate function. As an empirical study, we apply the single-index quantile regression model to Boston housing data. By considering different levels of quantile, we explore how the covariates, of either social or environmental nature, could have different effects on individuals targeting the low, the median, and the high end of the housing market.