# Depth-based Weighted Jackknife Empirical Likelihood for Non-smooth U-structure Equations

## TL;DR

This paper introduces a weighted jackknife empirical likelihood method to improve robustness against outliers in non-smooth U-structure equations, providing theoretical properties and demonstrating effectiveness through simulations and real data.

## Contribution

It proposes a novel weighted JEL approach that reduces outlier sensitivity and derives its asymptotic distribution, enhancing robustness in non-smooth U-statistic problems.

## Key findings

- WJEL converges to a scaled chi-square distribution.
- Self-normalized WJEL yields standard chi-square distribution.
- Simulation studies confirm robustness against outliers.

## Abstract

In many applications, parameters of interest are estimated by solving some non-smooth estimating equations with $U$-statistic structure. Jackknife empirical likelihood (JEL) approach can solve this problem efficiently by reducing the computation complexity of the empirical likelihood (EL) method. However, as EL, JEL suffers the sensitivity problem to outliers. In this paper, we propose a weighted jackknife empirical likelihood (WJEL) to tackle the above limitation of JEL. The proposed WJEL tilts the JEL function by assigning smaller weights to outliers. The asymptotic of the WJEL ratio statistic is derived. It converges in distribution to a multiple of a chi-square random variable. The multiplying constant depends on the weighting scheme. The self-normalized version of WJEL ratio does not require to know the constant and hence yields the standard chi-square distribution in the limit. Robustness of the proposed method is illustrated by simulation studies and one real data application.