A penalized version of the empirical likelihood ratio for the population mean

A penalized version of the empirical likelihood ratio test statistic for the population mean is proposed which may be computed even when this parameter does not belong to the convex hull of the data. Some theoretical results on the proposed test statistic are provided together with some guidelines on the choice of the penalization term. A double bootstrap procedure is also described which may be used for calibration when the sample size is small. The approach is illustrated by an example and a small simulation study.