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Expansions for the Conditional Density and Distribution of a Standard Estimate

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  • Christopher S. Withers

    (Formerly Industrial Research Ltd., Lower Hutt 6007, New Zealand)

Abstract

Conditioning is a very useful way of using correlated information to reduce the variability of an estimate. Conditioning an estimate on a correlated estimate, reduces its covariance, and so provides more precise inference than using an unconditioned estimate. Here we give expansions in powers of n − 1 / 2 for the conditional density and distribution of any multivariate standard estimate based on a sample of size n . Standard estimates include most estimates of interest, including smooth functions of sample means and other empirical estimates. We also show that a conditional estimate is not a standard estimate, so that Edgeworth-Cornish-Fisher expansions cannot be applied directly.

Suggested Citation

  • Christopher S. Withers, 2025. "Expansions for the Conditional Density and Distribution of a Standard Estimate," Stats, MDPI, vol. 8(4), pages 1-25, October.
    • Handle: RePEc:gam:jstats:v:8:y:2025:i:4:p:98-:d:1771266
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    References listed on IDEAS

    as
    1. Marcelo J. Moreira, 2003. "A Conditional Likelihood Ratio Test for Structural Models," Econometrica, Econometric Society, vol. 71(4), pages 1027-1048, July.
    2. Christopher Withers & Saralees Nadarajah, 2010. "Tilted Edgeworth expansions for asymptotically normal vectors," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 62(6), pages 1113-1142, December.
    3. Hansen, Bruce E, 1994. "Autoregressive Conditional Density Estimation," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 35(3), pages 705-730, August.
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