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 .
Given additional distributional information in the form of moment restrictions, kernel density and distribution function estimators with implied generalised empirical likelihood probabilities as weights achieve a reduction in variance due to the systematic use of this extra information. The particular interest here is the 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 , being potentially of use for diagnostic purposes, including tests of parametric assumptions on an error distribution, goodness-of-fit tests or tests of overidentifying moment restrictions. The paper gives conditions for the consistency and describes the asymptotic mean squared error properties of the kernel density and distribution estimators proposed in the paper. A simulation study evaluates the small sample performance of these estimators. Supplements provide analytic examples to illustrate situations where kernel weighting provides a reduction in variance together with proofs of the results in the paper .
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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
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testing parametric assumptions about the
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We give conditions under which such estimators are consistent, and describe their
The paper gives conditions for the consistency and describes the
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of the kernel density and distribution estimators proposed in the paper. A simulation study evaluates the small sample performance of these estimators. Supplements provide analytic examples to illustrate situations where kernel weighting
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, and a simulation study is conducted to evaluate small sample performance of the proposed estimators
together with proofs of the results in the paper