A review of recent advances in empirical likelihood

Abstract

Empirical likelihood is widely used in many statistical problems. In this article, we provide a review of the empirical likelihood method, due to its significant development in recent years. Since the introduction of empirical likelihood, variants of empirical likelihood have been proposed, and the applications of empirical likelihood in high dimensions have also been studied. It is necessary to summarize the new development of empirical likelihood. In this article, we give a review of the Bayesian empirical likelihood, the bias-corrected empirical likelihood, the jackknife empirical likelihood, the adjusted empirical likelihood, the extended empirical likelihood, the transformed empirical likelihood, the mean empirical likelihood, and the empirical likelihood with high dimensions. Finally, we have a brief survey of the computation and implementation for empirical likelihood methods.

Interval estimation in three-class receiver operating characteristic analysis: A fairly general approach based on the empirical likelihood

“…It is worth noting that the nonparametric log-likelihood ratio in (5), coincides with the log-likelihood ratio based on the (independent) binomial distributions of F1 (t 1 ), P2 (t 1 , t 2 ) and F3 (t 2 ). This provides a further justification for the result in Theorem 2.1 under the considered weak conditions, which do not even involve the continuity of F 1 , F 2 and F 3 .…”

“…Observe that, in (5), each of the three addenda is an EL pivot (with approximant distribution 𝜒 2 1 ) for inference about an unknown proportion or probability, estimated by its empirical counterpart; in a sample of independent and identically distributed observations, such empirical counterpart, multiplied by the sample size, is the realization of a binomial random variable.…”

“…Nowadays, EL methods have a wide range of applications in various research fields (see, Lazar 4 and Liu and Zhao, 5 for recent reviews), including the receiver operating characteristic (ROC) analysis for a three-class problem, which is commonly used to evaluate the ability of a diagnostic test (or biomarker) to distinguish three ordinal classes (e.g. benign, stage 1, stage 2) of a disease.…”

Robust meta gradient learning for high-dimensional data with noisy-label ignorance

“…However, when the dimension d of the covariate X diverges, it can be shown that the estimator in ( 8 ) is no longer consistent with the true parameter β 0 , and the estimation equation is no longer robust [ 16 ], which leads to the model prediction will have serious error accumulation and reduces the prediction accuracy [ 16 , 61 ]. Hence, to solve these problems, we introduce a class of penalty function mentioned in Liu et al (2023) [ 21 ] as follows, …”

“…Concretely, we begin by defining the γ -divergence and introducing a class of penalty function mentioned in Liu et al (2023) [ 21 ], which yields the objective function Eq (9) . Based on this, we derive the consistency and efficiency of our model parameter estimation (refer to Theorems 1 and 2), as well as the empirical risk upper bound of our model (refer to Theorem 3).…”