Small sample performance and calibration of the empirical likelihood method

The empirical likelihood method is a versatile approach for testing hypotheses and constructing confidence regions in a non-parametric setting. For testing the value of a vector mean, the empirical likelihood method offers the benefit of making no distributional assumptions beyond some mild moment conditions. However, in small samples or high dimensions the method is poorly calibrated, producing tests that generally have a much higher type I error than the nominal level, and it suffers from a limiting convex hull constraint. Methods to address the performance of the empirical likelihood method in the vector mean setting have been proposed by a number of authors. We briefly explore a variety of such methods, commenting on the abilities of the various methods to address the calibration and convex hull challenges. In particular, a recent contribution suggests supplementing the observed dataset with an artificial data point, thereby eliminating the convex hull issue. We examine the performance of this approach and describe a limitation of their method that we have discovered in settings when the sample size is relatively small compared with the dimension.;We propose a new modification of the extra data approach, adding two balanced points rather than one, and changing the method used to determine the location of the artificial points. This modification demonstrates markedly improved calibration in difficult examples, and also results in a small-sample connection between the modified empirical likelihood method and Hotelling's T-square test. Varying the location of the added data points creates a continuum of tests, ranging from essentially the unmodified empirical likelihood method to a scaled Hotelling's T-square test as the distance of the extra points from the sample mean increases. We investigate these consequences of the use and placement of artificial data points, exploring the accuracy of the chi-squared calibration and the power of the resulting test as the placement of the added points changes. Then we extend this method to multi-sample comparisons, where the new method demonstrates behavior comparable to that of multi-sample extensions of Hotelling's T-square test with corrections, and has the added advantage that it may be tuned by varying the placement of the artificial points.