Annals of Statistics

On the Density of Minimum Contrast Estimators

Ib M. Skovgaard

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Conditions for the existence of the density of a minimum contrast estimator in a parametric statistical family are given together with a formula for this density. The formula is exact if multiple local minima cannot occur; otherwise the formula is an exact expression for the point process of local minima of the contrast function. Although it is not in general feasible to compute the expression for the density, the formula can be used as a basis for further expansion of the large deviation type. When the estimate is sufficient, either in the original model or after conditioning on an approximate or exact ancillary, the formula simplifies drastically. In particular, it is shown how Barndorff-Nielsen's formula for the density of the maximum likelihood estimator given an ancillary statistic is derived from the formula given here. In this way the nature of Barndorff-Nielsen's formula as an asymptotic approximation and its appearance as an exact formula for certain cases are demonstrated.

Article information

Ann. Statist., Volume 18, Number 2 (1990), 779-789.

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62F12: Asymptotic properties of estimators
Secondary: 62E15: Exact distribution theory

Barndorff-Nielsen's formula conditional inference large deviation expansion minimum contrast estimator maximum likelihood estimator saddlepoint approximation


Skovgaard, Ib M. On the Density of Minimum Contrast Estimators. Ann. Statist. 18 (1990), no. 2, 779--789. doi:10.1214/aos/1176347625.

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