Derivation of a Broad Class of Consistent Estimates

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.