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Improved Density and Distribution Function Estimation

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  • Vitaliy Oryshchenko
  • Richard J. Smith

Abstract

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.

Suggested Citation

  • Vitaliy Oryshchenko & Richard J. Smith, 2017. "Improved Density and Distribution Function Estimation," Papers 1711.04793, arXiv.org, revised Jun 2018.
    • Handle: RePEc:arx:papers:1711.04793
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    References listed on IDEAS

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    1. Antoine, Bertille & Bonnal, Helene & Renault, Eric, 2007. "On the efficient use of the informational content of estimating equations: Implied probabilities and Euclidean empirical likelihood," Journal of Econometrics, Elsevier, vol. 138(2), pages 461-487, June.
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    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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