Generalised empirical likelihood-based kernel density estimation
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If additional information about the distribution of a random variable is available in the form of moment conditions, the weighted kernel density estimator constructed by replacing the uniform weights with the generalised empirical likelihood probabilities provides an improved approximation to the moment constraints. More importantly, a reduction in variance is achieved due to the systematic use of the extra information. Same approach can be used to estimate a density or distribution of certain functions of data and, possibly, of the unknown parameters, the latter being replaced by their generalised empirical likelihood estimates. A special case of interest is estimation of densities or distributions of (generalised) residuals in semi-parametric models defined by a finite number of moment restrictions. Such estimates are of great practical interest and can be used for diagnostic purposes, including testing parametric assumptions about the error distribution, goodness-of-fit, or overidentifying moment restrictions. We give conditions under which such estimators are consistent, and describe their asymptotic mean squared error properties. Analytic examples illustrate the situations where re-weighting provides a reduction in variance, and a simulation study is conducted to evaluate small sample performance of the proposed estimators.
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