Maximum Empirical Likelihood Estimation In A Heteroscedastic Linear Regression ModelWith Possibly Missing Responses

Anton Schίck ,Yilin Zhu

doi:10.4038/sljastats.v5i4.7791

Abstract

A heteroscedastic linear regression model is considered where responses are allowed to be missing at random and with the conditional variance modeled as a function of the mean response. Maximum empirical likelihood estimation is studied for an empirical likelihood with an increasing number of estimated constraints. The resulting estimator is shown to be asymptotically normal and can outperform the ordinary least squares estimator. DOI: http://dx.doi.org/10.4038/sljastats.v5i4.7791

Highlights

Consider the heteroscedastic linear regression model in which the response variable Y is linked to the q-dimensional covariate vector X by the formula
Y = θT Z + ε, where Z is m(X) for a known measurable function m from Rq into Rp, θ is an unknown vector in Rp, the error variable ε is conditionally centered, i.e., E[ |X] = 0, and its conditional variance σ2(X) = E[ 2|X]
For q = 1, a possible choice of m is m(x) = (1, x, x2, . . . , xp−1), x ∈ R, which corresponds to polynomial regression

Summary

Introduction

Schick (2013) has shown that one can construct an estimator that is asymptotically equivalent to the oracle weighted least squares estimator without constructing an estimator of the variance function σ2 He treated the case q = 1 with missing responses. His estimator maximizes the restriction of Sn to the random ball centered at the least squares estimator of radius C(log n/n)1/2 for some constant C, θS = He obtained the asymptotic equivalence of this estimator and the oracle weighted least squares estimator (by establishing the expansion (1.1) with θS in place of θW ) under a growth condition on rn and mild assumptions on the functions vr.

Simulations

Comments and Remarks

Proof of Theorem 1