Consistency, efficiency and robustness of conditional disparity methods

TL;DR

This paper extends minimum-disparity estimators for conditional regression models, demonstrating their consistency, asymptotic normality, robustness, and bias correction methods, including bootstrap techniques, across models with mixed data types.

Contribution

It introduces a broad class of consistent, robust conditional disparity estimators with kernel density methods, addressing bias and providing empirical bias correction techniques.

Findings

Establishes consistency and asymptotic normality of the estimators.

Demonstrates robustness of the estimators against model misspecification.

Shows bootstrap methods effectively reduce bias and improve confidence intervals.

Abstract

This paper considers extensions of minimum-disparity estimators to the problem of estimating parameters in a regression model that is conditionally specified; that is where a parametric model describes the distribution of a response $y$ conditional on covariates $x$ but does not specify the distribution of $x$. We define these estimators by estimating a non-parametric conditional density estimates and minimizing a disparity between this estimate and the parametric model averaged over values of $x$. The consistency and asymptotic normality of such estimators is demonstrated for a broad class of models in which response and covariate vectors can take both discrete and continuous values and incorportates a wide set of choices for kernel-based conditional density estimation. It also establishes the robustness of these estimators for a broad class of disparities. As has been observed in Tamura and Boos (J. Amer. Statist. Assoc. 81 (1986) 223--229), minimum disparity estimators incorporating kernel density estimates of more than one dimension can result in an asymptotic bias that is larger that $n^{-1/2}$ and we characterize a similar bias in our results and show that in specialized cases it can be eliminated by appropriately centering the kernel density estimate. We also demonstrate empirically that bootstrap methods can be employed to reduce this bias and to provide robust confidence intervals. In order to demonstrate these results, we establish a set of $L_1$-consistency results for kernel-based estimates of centered conditional densities.