Empirical likelihood confidence regions for the parameters of a two phases nonlinear model with and without missing response data

TL;DR

This paper develops empirical likelihood confidence regions for parameters in two-phase nonlinear models, including cases with missing data, demonstrating their asymptotic chi-squared distribution and validating through simulations.

Contribution

It introduces three empirical likelihood methods for constructing confidence regions in nonlinear models with missing data, extending Wilk's theorem nonparametrically.

Findings

Empirical likelihood ratios follow asymptotic chi-squared distribution.

The proposed methods achieve good coverage probabilities.

Simulation results confirm the effectiveness of the approaches.

Abstract

In this paper, we use the empirical likelihood method to construct the confidence regions for the difference between the parameters of a two-phases nonlinear model with random design. We show that the empirical likelihood ratio has an asymptotic chi-squared distribution. The result is a nonparametric version of Wilk's theorem. Empirical likelihood method is also used to construct the confidence regions for the difference between the parameters of a two-phases nonlinear model with response variables missing at randoms (MAR). In order to construct the confidence regions of the parameter in question, we propose three empirical likelihood statistics : Empirical likelihood based on complete-case data, weighted empiri- cal likelihood and empirical likelihood with imputed values. We prove that all three empirical likelihood ratios have asymptotically chi-squared distributions. The effectiveness of the proposed approaches in aspects of coverage probability and interval length is demonstrated by a Monte-Carlo simulations.