Han-Ying Liang

Empirical Likelihood for a Heteroscedastic Partial Linear Model

Consider the heteroscedastic regression model , where , β is a p × 1 column vector of unknown parameter, (X i , T i , Z i ) are random design points, Y i are the response variables, g(·) is an unknown function defined on the closed interval [0, 1], {e i , ℱ i } is a sequence of martingale differences. When f is known and unknown cases, we propose the empirical log-likelihood ratio statistics for the parameter β. For each case, a nonparametric version of Wilks theorem is derived. The results are then used to construct confidence regions of the parameter. Simulation study shows that the empirical likelihood method performs better than a normal approximation-based approach.

Asymptotic Normality of Estimators in Heteroscedastic Semi-Parametric Model with Strong Mixing Errors

Consider the heteroscedastic semi-parametric model y i = x i β +g(t i ) + σ i e i (1 ≤ i ≤ n), where , the design points (x i , t i , u i ) are known and non random, g(·) and f(·) are unknown functions defined on closed interval [0, 1], and the random errors e i are assumed to be a sequence of stationary α-mixing random variables with mean zero. Under appropriate conditions, we study the asymptotic normality of least-squares estimator and two weighted least-squares estimators of β. Also, the asymptotic normality of the estimators of g(·) and f(·) is considered. Finite sample behavior of the estimators is investigated via simulations as well.

Asymptotic Properties of Conditional Quantile Estimator Under Left-Truncated and α-Mixing Conditions

In this article, we establish strong uniform convergence and asymptotic normality of estimators of conditional quantile and conditional distribution function for a left truncated model when the data exhibit some kind of dependence. It is assumed that the observations form a stationary α-mixing sequence. The results of Lemdani et al. (2009) are relaxed from the i.i.d. assumption to α-mixing setting. Finite sample behavior of the estimators is investigated via simulations as well.

Asymptotic Normality for Regression Function Estimate Under Truncation and α-Mixing Conditions

In this article we establish pointwise asymptotic normality of nonparametric kernel estimator of regression function for a left truncation model. It is assumed that the lifetime observations with multivariate covariates form a stationary α-mixing sequence. Also, the asymptotic normality of the estimation of the covariables density is considered. As a by-product, we obtain a uniform weak convergence rate for the product-limit estimator of the lifetime and truncated distributions under dependence, which is interesting independently. Finite sample behavior of the estimator of the regression function is investigated as well.

Convergence rate of wavelet density estimator with data missing randomly when covariables are present

ABSTRACT In this article, we study global L2 error of non linear wavelet estimator of density in the Besov space Bspq for missing data model when covariables are present and prove that the estimator can achieve the optimal rate of convergence, which is similar to the result studied by Donoho et al. (1996) in complete independent data case with term-by-term thresholding of the empirical wavelet coefficients. Finite-sample behavior of the proposed estimator is explored via simulations.

Berry–Esseen type bound of conditional mode estimation under truncation and strong mixing assumptions

ABSTRACT In order to investigate the convergence rate of the asymptotic normality for the estimator of the conditional mode function for the left-truncation model, we derive a Berry–Esseen type bound of the estimator when the lifetime observations with multivariate covariates form a stationary α-mixing sequence. The finite sample performance of the estimator of the conditional mode function is explored through simulations.

Berry–Esseen type bounds in heteroscedastic errors-in-variables model

ABSTRACT Consider the heteroscedastic partially linear errors-in-variables (EV) model yi = xiβ + g(ti) + εi, ξi = xi + μi (1 ⩽ i ⩽ n), where εi = σiei are random errors with mean zero, σ2i = f(ui), (xi, ti, ui) are non random design points, xi are observed with measurement errors μi. When f( · ) is known, we derive the Berry–Esseen type bounds for estimators of β and g( · ) under {ei,u20091 ⩽ i ⩽ n} is a sequence of stationary α-mixing random variables, when f( · ) is unknown, the Berry–Esseen type bounds for estimators of β, g( · ), and f( · ) are discussed under independent errors.

Empirical Likelihood for a Heteroscedastic Partial Linear Errors-in-Variables Model

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 heteroscedastic partially linear errors-in-variables model with martingale difference errors. 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.