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 when nuisance parameters are present. This paper presents two approximations to the profile empirical likelihood function for a scalar parameter of inter...

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Bibliographic Details
Published in:Biometrika 2001-06, Vol.88 (2), p.337-351
Main Authors: DiCiccio, Thomas J., Monti, Anna Clara
Format: Article
Language:English
Subjects:
Approximation
Asymptotic expansion
Bootstrap calibration
Bootstrap resampling
Calibration
Confidence interval
Confidence intervals
Data lines
Estimators
Exact sciences and technology
Iterative methods
Mathematical foundations
Mathematical independent variables
Mathematics
Nonlinear equations
Nonparametric inference
Nonparametric likelihood
Numerical differentiation
Parameter transformation
Parametric inference
Probability and statistics
Scalars
Sciences and techniques of general use
Skewness
Statistics
Studentised statistic
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Summary: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.
ISSN:0006-3444
1464-3510
DOI:10.1093/biomet/88.2.337