Empirical likelihood confidence regions of the parameters in a partially single-index varying-coefficient model

1 Introduction

Consider a partially single-index varying-coefficient model of the form

where (U, X, Z) ∈ R^p × R^d × R^q is a vector of covariates, Y is the response variable, β₀ is a p × 1 vector of unknown parameters, θ₀ is a q × 1 vector of regression coefficient, g₀(⋅) is a d × 1 vector of unknown functions and ε is a random error with E(ε | U, X, Z) = 0 and Var(ε| U, X, Z) = σ². Assume that ε and (U, X, Z) are independent. In order to make sure the identifiability, it is often assumed that ∥β₀∥ = 1, where ∥⋅∥ denotes the Euclidean metric.

Feng and Xue[1] considered the problem of model detection and estimation for single-index varying-coefficient model, they identified the true model structure and obtained a new semiparametric model, which is partially linear single-index varying-coefficient model. As we all know, the optimal parametric estimation rate is n^−1/2 and the optimal nonparametric estimation rate is n^−2/5, if we treat a parametric component as a nonparametric component, the problems of data overfitting and efficiency loss will occur.

Model (1.1), may include crossproduct terms of some components of X and Z, is easily interpreted in real applications because it has the features of the partial linear model, the single-index model and the varying-coefficient model, which make the model more general. Model (1.1) takes many other regression models as special cases, such as linear model, partial linear model, varying-coefficient model, single-index model, partial linear single-index model, single-index varying-coefficient model, etc. The linear component θ0τZ provides a simple summary of covariates effects which are of the main scientific interest. The index β0τU enables us to simplify the treatment of the multiple auxiliary variables, and the functions g₀(⋅)s enrich model flexibility. It is well known that in order to construct the confidence region of (β₀, θ₀) by using the normal approximation method, it is necessary to construct the embedded estimation of the asymptotic variance of the corresponding estimator, which includes the estimation of parametric and non-parametric components. The empirical likelihood method avoids this shortcoming and its structure does not include estimation of parameter (β₀, θ₀). In this paper, we can construct an empirical likelihood ratio function for (β₀, θ₀) by assuming g₀(⋅) and its derivative ġ₀(⋅) to be known functions.

As far as we know, there is not much literature on this model by using empirical likelihood method, although it has been applied to varieties of models. In this paper, we consider the problem of a method of constructing confidence regions for (β₀, θ₀), since the empirical likelihood method, which is introduced by Owen[2, 3], has many advantages. For example, it does not require the construction of a pivotal quantity, and it does not impose a priori constraint on the shape of the region. Owen[4] proved the empirical log-likelihood ratio is asymptotically a standard chi-square variable when he applied the empirical likelihood to linear regression model, so that it can be applied to constructing the confidence region of the regression parameter. There are studies related to empirical likelihood, such as Wang and Rao[5], Wang, Linton and Härdle[6], Xue and Zhu[7, 8, 9], Zhu and Xue[10], You and Zhou[11], Qin and Zhang[12], Stute, Xue and Zhang[13], Xue[14, 15], Wang et al.[16], Huang and Zhang[17], Wang and Xue[18], Lian[19], Xiao[20], Zhou, Zhao and Wang[21], Fang, Liu and Lu[22], Arteaga-Molina and Rodriguez-Poo[23], among others.

The rest of this article is organized as follows. In Section 2, we give an estimated empirical likelihood ratio, and investigate the asymptotic properties of the proposed estimators. In Section 3, we give an adjusted empirical log-likelihood and derive its asymptotic distribution. Section 4 reports a simulation study. Proofs of theorems and lemmas are postponed in Appendix A and Appendix B, respectively.

2 Main results

3 Adjusted empirical likelihood

When we use Theorem (2.1) to construct confidence regions of (β, θ), the weights w_is need to be estimated, the accuracy of confidence region is decreased. Let ρ(β₀, θ₀) = (p + q)/tr{V₀(β₀, θ₀)}, where V₀(β₀, θ₀) is defined in Theorem (2.1). By Rao and Scott[27], the distribution of ρ(β0,θ0)∑i=1p+qwiχ1,i2 can be approximated by χp+q2 , which is a standard chi-square distribution with p + q degrees of freedom. Therefore, an improved Rao-Scott adjusted empirical log-likelihood ratio can be defined as

However, the accuracy of this approximation depends on the w_is. Xue and Wang[28] proposed another adjusted empirical log-likelihood. By using an approximate result in the above, the adjustment technique is developed by Wang and Rao[5]. Note that ρ̂(β, θ) can be written as

where A⁻ represents a generalized inverse of matrix A. By examining the asymptotic expansion of −2logL̂(β, θ), similar to Xue and Wang[28], we can define an adjustment factor

where Σ^(β,θ)={∑i=1nη^i(β,θ)}{∑i=1nη^i(β,θ)}τ . The adjusted empirical log-likelihood ratio is defined by

The following theorem shows that the adjusted empirical log-likelihood ratio is asymptotically distributed as standard chi-square.

Invoking Theorem (3.1), l̂(β, θ) can be used to construct an approximate confidence region for (β₀, θ₀). Thereby, we can obtain the confidence region of (β, θ)

where P(χp+q2≤c1−α)=1−α .

To apply Theorem (3.1) to construct a confidence region for (β₀, θ₀), we only need to estimate the adjustment factor r̂(β, θ) by replacing (β, θ) by (β̂, θ̂). The value of −2logL̂(β, θ) is not depend on the estimation of (β, θ). In practice, we can calculate the numerical value of −2logL̂(β, θ) by using the package in R(see ‘emplik’, http://cran.r-project.org/web/packages/emplik/).

4 Simulation study

Consider the regression model

where β0=(2/2,2/2)τ , θ₀ = (2, 1.2)^τ, ε_i ∼ N(0, 0.3²). The sample {U_i = (U_i1, U_i2)^τ; 1 ≤ i ≤ n} was generated from a bivariate uniform distribution [−1, 1]² with independent components, X_i1, X_i2 were generated from a normal distribution N(0, 0.8²) and {Z_i = (Z_i1, Z_i2)^τ; 1 ≤ i ≤ n} was generated from a bivariate normal distribution N(0, Σ) with Cov(Z_ik, Z_il) = 2 × 0.5^|k−l|, k, l = 1, 2. In model (4.1), the coefficient functions are g₁(t) = sin(t − 2) and g₂(t) = 1 − t².

We used a Epanechnikov kernel K(t) = 0.75(1 − t²)₊ and took the weight function w(t)=I[−2,2](t) . The bandwidths ĥ = ĥ₁ n^−1/25(log n)^−1/2 and ĥ₁ = ĥ_opt respectively, where ĥ_opt was an optimal bandwidth by using the generalized cross validation(GCV). It’s not hard to see that the bandwidth ĥ satisfies condition (C4).

From Table 1, we can see that the probability of coverage increases with n until it approaches the theoretical value $0.95$.

5 Appendices