The Annals of Mathematical Statistics

Derivation of a Broad Class of Consistent Estimates

R. C. Davis

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Abstract

Given a chance vector $\mathbf{X}$ with distribution function $F(\mathbf{X}, \theta_T)$, where $\mathbf{theta}_T$ denotes the true unknown parameter vector, a broad class of estimates of $\mathbf{theta}_T$ is derived which is shown to be identical with the class of all consistent estimates of $\mathbf{\theta}_T$. A sub-class is obtained each member of which has the following properties: a.) Its construction depends upon the solution of an equation involving a single vector function of the parameter vector $\theta$ and the members of a sequence $\{\mathbf{X}_n\}$ of independent and identically distributed chance vectors; b.) the estimate so obtained converges almost certainly to $\mathbf{\theta}_T$; c.) it is a symmetric function of the members of the sequence $\{\mathbf{X}_n\}$. In order to obtain this subclass it is postulated that a function of $\mathbf{X}$ and $\mathbf{\theta}$ exists (continuous in $\mathbf{\theta}$ for a certain neighborhood of the true parameter $\mathbf{\theta}_T$ and existing for each $\mathbf{X}$ in a subset of the sample space) which satisfies a Lipschitz condition in $\mathbf{\theta}$. In particular if a density function $f(\mathbf{X}, \theta_T)$ exists satisfying certain conditions, the consistency of the maximum likelihood estimate can be established under regularity conditions quite different from those usually assumed [1]. This is not to be interpreted as a weakening of the usual regularity conditions but rather as an extension of the class of consistent likelihood estimates obtained under the usual regularity conditions.

Article information

Source
Ann. Math. Statist., Volume 21, Number 3 (1950), 425-431.

Dates
First available in Project Euclid: 28 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177729801

Digital Object Identifier
doi:10.1214/aoms/1177729801

Mathematical Reviews number (MathSciNet)
MR36481

Zentralblatt MATH identifier
0039.15001

JSTOR
links.jstor.org

Citation

Davis, R. C. Derivation of a Broad Class of Consistent Estimates. Ann. Math. Statist. 21 (1950), no. 3, 425--431. doi:10.1214/aoms/1177729801. https://projecteuclid.org/euclid.aoms/1177729801


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