Depth-based weighted jackknife empirical likelihood for non-smooth U-structure equations

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. maximization problem of the EL with nonlinear constraints to the simple case of EL on the mean of jackknife pseudo-values, which is very effective in handling one and two-sample U -statistics. Since then, it has attracted strong interests in a wide range of fields due to its efficiency, and many papers are devoted to the investigation of the method, for example, Liu, Xia and Zhou ([20]), Peng ([26]), ([30]) and so on. In many nonparametric and semiparametric approaches, such as the Gini correlation, quantile regression and rank regression, the parameters of interest are estimated by solving equations with U -statistic structure instead of directly by U -statistics. Thus, the JEL in Jing, Yuan and Zhou ([11]) can not be applied directly. Li, Xu and Zhou ([17]) extended the JEL to the more complicated but more general situation. The Wilks' theorems are established even for the situation in which nuisance parameters are involved.As the EL method is sensitive to outliers and the EL confidence regions may be greatly lengthened in the directions of the outliers (Owen[25], Tsao and Zhou[36]), the JEL method with equation constraints is sensitive to outliers. That is, the JEL method is not robust. For the EL approach, a number of methods have been proposed to achieve robustness, see Wu ([42]), Glenn and Zhao ([8]), Jiang et al. ([12]). Those robust empirical likelihood (REL) methods tilt the EL function by assigning smaller weights to outliers, which yield a more robust estimator and confidence region. Jiang et al. ([12]) linked the depth-based weighted empirical likelihood (WEL) with general estimating equations and produced a robust estimation of parameters. They constructed weights based on a depth function although it is not the spatial depth as they claimed. Data depth provides a centre-outward ordering of multi-dimensional data. Points deep inside the data are assigned with a high depth and those on the outskirts with a lower depth. In the literature, depth functions have been extensively studied, for example, Mahalanobis depth ([28]), ...

A review of recent advances in empirical likelihood

“…Cheng et al (2018) added two artificial points to the original pseudo‐value data set and developed the balanced augmented JEL. To solve the sensitivity problems with outliers, Sang et al (2020) derived the weighted JEL by assigning smaller weights to outliers. From the perspective of the jackknife empirical likelihood, Matsushita and Otsu (2021) discussed the nonstandard asymptotic frameworks, such as small‐bandwidth asymptotics and sparse‐network asymptotics.…”