Empirical Likelihood

Empirical likelihood is a technique for forming hypothesis tests and conndence sets based on nonparametric likelihood ratios. Many properties of parametric likelihood ratio functions have nonparametric parallels. The main one is that there is a nonparametric version of Wilks's famous result wherein an asymptotic chisquare distribution holds for the log likelihood ratio. The nonparametric version has the advantage of holding under very weak conditions. In particular, the data do not have to follow a parametric distribution for the chisquare result to hold for nonparametric likelihood ratios. Empirical likelihood ratio inferences are of comparable accuracy to those based on the delta method, the jackknife and the simpler bootstrap methods. Each method has its advantages. Here we list a few of empirical likelihood's advantages: When constructing conndence regions for two or more parameters of interest, empirical likelihood gives a data determined shape for the conndence region. When constraints are known to hold among the parameters of interest, they can be imposed numerically. A Bartlett correction applies to empirical likelihood, providing a simple way to increase the accuracy of inferences. As in some, but not all of the other methods, all points in an empirical likelihood conndence region obey the usual range restrictions: variances are nonnegative, probabilities are in 0; 1] and correlations are in ?1; 1]. Sometimes empirical likelihood methods require less computation than the alternatives, sometimes more. At rst nonparametric likelihood ratios may seem paradoxical, but they are deened below by taking very literally the notion that likelihood is the probability of observing the actual data values at hand.