Sequential Linearization of Empirical Likelihood Constraints with Application to U-Statistics

Empirical likelihood for a mean is straightforward to compute, but for nonlinear statistics significant computational difficulties arise because of the presence of nonlinear constraints in the underlying optimization problem. It is certainly the case that these difficulties can be overcome with sufficient time, care, and programming effort. However, they do make it difficult to write general software for implementing empirical likelihood, and therefore these difficulties are likely to hinder the widespread use of empirical likelihood in applied work. The purpose of this article is to suggest an approximate approach that sidesteps the difficult computational issues. The basic idea, which may be described as “sequential linearization of constraints,” is a very simple one, but we believe it could have significant ramifications for the implementation and practical use of empirical likelihood methodology. One application of the linearization approach, which we consider in this article, is to the problem of constructing empirical likelihood for U-statistics. However, the sequential linearization idea can be applied in the empirical likelihood context to a broad range of smooth statistical functional.