Approximations to the profile empirical likelihood function for a scalar parameter in the context of M-estimation

Empirical likelihood possesses many of the important properties of genuine parametric likelihood, but it is computationally burdensome, especially whe n nuisance parameters are present. This paper presents two approximations t o the profile empirical likelihood function for a scalar parameter of inter est in the context of M-estimation; the simpler approximation is based on t he third derivative of the profile log empirical likelihood function at its maximising point, while the more accurate approximation involves both the third and fourth derivatives. Formulae are given for these higher-order der ivatives that can be evaluated using ordinary matrix operations, so computa tion of the approximations is very easy. The accuracy of the approximations is demonstrated in several numerical examples. The computational simplicit y of the approximations makes it feasible to use them in conjunction with b ootstrap calibration for constructing accurate confidence intervals and lim its. The derivatives are also helpful for exploring the shape of the profil e log empirical likelihood function and for determining suitable parameteri sations for studentised statistics.