Empirical likelihood for single-index models

Abstract

The empirical likelihood method is especially useful for constructing confidence intervals or regions of the parameter of interest. This method has been extensively applied to linear regression and generalized linear regression models. In this paper, the empirical likelihood method for single-index regression models is studied. An estimated empirical log-likelihood approach to construct the confidence region of the regression parameter is developed. An adjusted empirical log-likelihood ratio is proved to be asymptotically standard chi-square. A simulation study indicates that compared with a normal approximation-based approach, the proposed method described herein works better in terms of coverage probabilities and areas (lengths) of confidence regions (intervals).

Empirical likelihood in some nonparametric and semiparametric models

“…For identifiability purposes, we typically assume that β = 1 with its first nonzero element being positive, where · denotes the Euclidean norm. Xue and Zhu (2006) considered the following approach to construct confidence region of β by empirical likelihood. Suppose that the recorded data {(X i , Y i ), 1 ≤ i ≤ n} are generated by the model (1), this is…”

Robust exponential squared loss-based variable selection for high-dimensional single-index varying-coefficient model

“…For the sake of identifiability, we assume that ||β|| = 1, the first component of β is positive, and g(x) cannot be the form as g(x) = α T xβ T x + γ T x + c, where || · || denotes the Euclidean metric, α, γ ∈ R p , c ∈ R are constants, and α and β are not parallel to each other (see [8,35]). It is important to emphasize that model (1.1) is flexible enough to cover many important models such as the standard single-index model (see [11,26,30,32,36]) and the varying coefficient model (see [2,7,12,29,37]). Thus, model (1.1) is easily interpreted in real applications because it has the features of both the single-index model and the varying-coefficient model.…”

Empirical likelihood methods based on influence functions

“…This so-called plugged-in empirical likelihood method has been widely applied under various semiparametric models by much literature. For example, [14,17] on partially linear models, [11] on Cox models, [10,5] on censored linear models, [12] on censored quantile models, [21,26] on signle-index models, [6,24] on linear transformation models, [18,19] on linear models with missing responses, among many others.…”

“…In many cases, instead of converging to a chi-squared distribution, the log-empirical likelihood ratios with estimators being plugged in converge to a weighted sum of several independent chi-squared distributions, c.f. [10,18,19,5,12,21,26,6,24], etc. The weights, depending on the limiting variances of the estimators for parameters of interest, are unknown.…”