## Annals of Statistics

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

### On the Density of Minimum Contrast Estimators

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176347625

**Digital Object Identifier**

doi:10.1214/aos/1176347625

**Mathematical Reviews number (MathSciNet)**

MR1056336

**Zentralblatt MATH identifier**

0709.62029

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62F12: Asymptotic properties of estimators

Secondary: 62E15: Exact distribution theory

**Keywords**

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

#### Citation

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