Weighted empirical likelihood estimates and their robustness properties

Maximum likelihood methods are by far the most popular methods for deriving statistical estimators. However, parametric likelihoods require distributional specifications. The empirical likelihood is a nonparametric likelihood function that does not require such distributional assumptions, but is otherwise analogous to its parametric counterpart. Both likelihoods assume that the random variables are independent with a common distribution. A nonparametric likelihood function for data that are independent, but not necessarily identically distributed is introduced. The contaminated normal density is used to compare the robustness properties of weighted empirical likelihood estimators to empirical likelihood estimators. It is shown that as the contamination level of the sample increases, the root mean squared error of the empirical likelihood estimator for the mean increases. Conversely, the root mean squared error of the weighted empirical likelihood estimator for the mean remains closer to the theoretical root mean squared error.