© 2001 by Biometrika Trust
Approximations to the profile empirical likelihood function for a scalar parameter in the context of M-estimation
1 Department of Social Statistics, Cornell University, Ithaca, New York 14853, U.S.Atjd9{at}cornell.edu 2 Faculty of Economics, University of Sannio, Piazza Guerrazzi 1, 82100 Benevento, Italy.acmonti{at}unisannio.it
Empirical likelihood possesses many of the important properties of genuine parametric likelihood, but it is computationally burdensome, especially when nuisance parameters are present. This paper presents two approximations to the profile empirical likelihood function for a scalar parameter of interest in the context of M-estimation; the simpler approximation is based on the 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 derivatives that can be evaluated using ordinary matrix operations, so computation of the approximations is very easy. The accuracy of the approximations is demonstrated in several numerical examples. The computational simplicity of the approximations makes it feasible to use them in conjunction with bootstrap calibration for constructing accurate confidence intervals and limits. The derivatives are also helpful for exploring the shape of the profile log empirical likelihood function and for determining suitable parameterisations for studentised statistics.
Key Words: Asymptotic expansion; Bootstrap calibration; Confidence interval; Nonparametric likelihood; Parameter transformation; Skewness; Studentised statistic
Received October 1999. Revised August 2000
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