Zhensheng Huang

Profile empirical-likelihood inferences for the single-index-coefficient regression model

This article deals with a new profile empirical-likelihood inference for a class of frequently used single-index-coefficient regression models (SICRM), which were proposed by Xia and Li (J. Am. Stat. Assoc. 94:1275–1285, 1999a). Applying the empirical likelihood method (Owen in Biometrika 75:237–249, 1988), a new estimated empirical log-likelihood ratio statistic for the index parameter of the SICRM is proposed. To increase the accuracy of the confidence region, a new profile empirical likelihood for each component of the relevant parameter is obtained by using maximum empirical likelihood estimators (MELE) based on a new and simple estimating equation for the parameters in the SICRM. Hence, the empirical likelihood confidence interval for each component is investigated. Furthermore, corrected empirical likelihoods for functional components are also considered. The resulting statistics are shown to be asymptotically standard chi-squared distributed. Simulation studies are undertaken to assess the finite sample performance of our method. A study of real data is also reported.

TESTS FOR VARYING-COEFFICIENT PARTS ON VARYING-COEFFICIENT SINGLE-INDEX MODEL

To study the relationship between the levels of chemical pol- lutants and the number of daily total hospital admissions for respiratory diseases and to find the eect of temperature/relative humidity on the admission number, Wong et al. (17) introduced the varying-coecient single-index model (VCSIM). As pointed out, it is a popular multivari- ate nonparametric fitting technique. However, the tests of the model have not been very well developed. In this paper, based on the estima- tors obtained by the local linear technique, the average method and the one-step back-fitting technique in the VCSIM, the generalized likelihood ratio (GLR) tests for varying-coecient parts on the VCSIM are estab- lished. Under the null hypotheses the new proposed GLR tests follow the ´ 2 -distribution asymptotically with scale constant and degree of freedom independent of the nuisance parameters, known as Wilks phenomenon. Simulations are conducted to evaluate the test procedure empirically. A real example is used to illustrate the performance of the testing approach.

Efficient penalized estimating method in the partially varying-coefficient single-index model

In this paper, penalized estimating equations are proposed to estimate the index parametric components, which is of primary interest, in the partially varying-coefficient single-index models (PVCSIMs). Although some procedures have been developed to estimate the index parameter in PVCSIM, the problem of how to conduct variable selection for the index in such models has not been addressed to date. To solve this problem, we propose a class of efficient penalized estimating equations, which combine the smoothly clipped absolute deviation (SCAD) penalty and a stepwise estimation method. The proposed method can simultaneously select significant variables in the index and estimate the nonzero smooth coefficient parameters. Under suitable conditions, we establish the theoretical properties of our penalized estimating procedure, including the oracle properties and the asymptotic normality for the resulting penalized estimation. We evaluate the performance of the proposed method by using Monte Carlo simulations and the application to a real dataset.

Empirical Likelihood for Semiparametric Varying-Coefficient Heteroscedastic Partially Linear Errors-in-Variables Models

The purpose of this article is to use the empirical likelihood method to study construction of the confidence region for the parameter of interest in semiparametric varying-coefficient heteroscedastic partially linear errors-in-variables models. When the variance functions of the errors are known or unknown, we propose the empirical log-likelihood ratio statistics for the parameter of interest. For each case, a nonparametric version of Wilks’ theorem is derived. The results are then used to construct confidence regions of the parameter. A simulation study is carried out to assess the performance of the empirical likelihood method.

Efficient empirical-likelihood-based inferences for the single-index model

This article proposes the efficient empirical-likelihood-based inferences for the single component of the parameter and the link function in the single-index model. Unlike the existing empirical likelihood procedures for the single-index model, the proposed profile empirical likelihood for the parameter is constructed by using some components of the maximum empirical likelihood estimator (MELE) based on a semiparametric efficient score. The empirical-likelihood-based inference for the link function is also considered. The resulting statistics are proved to follow a standard chi-squared limiting distribution. Simulation studies are undertaken to assess the finite sample performance of the proposed confidence intervals. An application to real data set is illustrated.

Adaptive profile-empirical-likelihood inferences for generalized single-index models

We study generalized single-index models and propose an efficient equation for estimating the index parameter and unknown link function, deriving a quasi-likelihood-based maximum empirical likelihood estimator (QLMELE) of the index parameter. We then establish an efficient confidence region for any components of the index parameter using an adaptive empirical likelihood method. A pointwise confidence interval for the unknown link function is also established using the QLMELE. Compared with the normal approximation proposed by Cui et al. [Ann Stat. 39 (2011) 1658], our approach is more attractive not only theoretically but also empirically. Simulation studies demonstrate that the proposed method provides smaller confidence intervals than those based on the normal approximation method subject to the same coverage probabilities. Hence, the proposed empirical likelihood is preferable to the normal approximation method because of the complicated covariance estimation. An application to a real data set is also illustrated.

Empirical likelihood for partially time-varying coefficient models with dependent observations

In this paper, we apply the empirical likelihood method to study the partially time-varying coefficient models with a random design and a fixed design under dependent assumptions. A nonparametric version of Wilks’ theorem is derived for the fixed-design case. For the random-design case, it is proved that the empirical log-likelihood ratio of the regression parameters admits a limiting distribution with a weighted sum of independent chi-squared distributions. In order that Wilks’ phenomenon holds, we propose an adjusted empirical log-likelihood (ADEL) ratio for the regression parameters. The ADEL is shown to have a standard chi-squared limiting distribution. Simulation studies are undertaken to indicate that the proposed methods work better than the normal approximation-based approach.

Partially linear single-index beta regression model and score test

An important model in handling the multivariate data is the partially linear single-index regression model with a very flexible distribution-beta distribution, which is commonly used to model data restricted to some open intervals on the line. In this paper, the score test is extended to the partially linear single-index beta regression model. The penalized likelihood estimation based on P-spline is proposed. Based on the estimation, the score test statistics about varying dispersion parameter is given. Its asymptotical property is investigated. Both simulated examples are used to illustrate our proposed methods.

Statistical estimation in partially linear single-index models with error-prone linear covariates

This article considers a class of partially linear single-index models when some linear covariates are not observed, but their ancillary variables are available. This model can avoid the ‘curse of dimensionality’ in multivariate nonparametric regressions, and it contains many existing statistical models such as the partially linear model (Engle, R.F., Granger, W. J., Rice, J., and Weiss, A. (1986), ‘Semiparametric Estimates of the Relation Between Weather and Electricity Sales’, Journal of The American Statistical Association, 80, 310–319), the single-index model (Härdle, W., Hall, P., and Ichimura, H. (1993), ‘Optimal Smoothing in Single-Index Models’, The Annals of Statistics, 21, 157–178), the partially linear errors-in-variables model (Liang, H., Härdle, W., and Carroll, R.J. (1999), ‘Estimation in a Semi-parametric Partially Linear Errors-in-Variables Model’, The Annals of Statistics, 27, 1519–1535), the partially linear single-index measurement error model (Liang, H., and Wang, N. (2005), ‘Partially Linear Single-Index Measurement Error Models’, Statistica Sinica, 15, 99–116), and so on as special examples. In this article, an estimation procedure for the unknowns of the proposed models is proposed, and asymptotic properties of the corresponding estimators are derived. Finite sample performance of the proposed methodology is assessed by Monte Carlo simulation studies. A real example is also given to illustrate the proposed procedures.

Empirical likelihood for generalized partially linear varying-coefficient models

Generalized partially linear varying-coefficient models (GPLVCM) are frequently used in statistical modeling. However, the statistical inference of the GPLVCM, such as confidence region/interval construction, has not been very well developed. In this article, empirical likelihood-based inference for the parametric components in the GPLVCM is investigated. Based on the local linear estimators of the GPLVCM, an estimated empirical likelihood-based statistic is proposed. We show that the resulting statistic is asymptotically non-standard chi-squared. By the proposed empirical likelihood method, the confidence regions for the parametric components are constructed. In addition, when some components of the parameter are of particular interest, the construction of their confidence intervals is also considered. A simulation study is undertaken to compare the empirical likelihood and the other existing methods in terms of coverage accuracies and average lengths. The proposed method is applied to a real example.