On Regular Best Asymptotically Normal Estimates

Abstract

This study was initiated in connection with estimating parameters involved in a certain stochastic process of population growth. Because of the nature of distribution functions arising in such studies, the usual methods of estimation result in formulas which are so complex that it is difficult, if not impossible, to obtain explicit solutions for the estimates of the parameters. Investigation of the problem led to an extension of the method of best asymptotically normal estimates developed by Neyman [1]. The estimates derived are termed regular best asymptotically normal estimates (RBAN estimates). This extension can be applied to other problems. In [1], Neyman considers a whole class of estimates which possess the properties of consistency, of asymptotic normality, and of asymptotic efficiency, and he provides estimates having these asymptotic properties for the case of multinomial distributions. His method is extended in the present paper to a more general case in which random vectors are dealt with. Such an extension was considered by Barankin and Gurland [2], who studied a large class of estimates and showed that if the distributions involved are members of Koopman's family, it is still possible to reach the Cramer-Rao lower bound. The purposes of the present paper are to discuss a subclass of the estimates considered by Barankin and Gurland and to present simple methods of generating such estimates. The estimates discussed are based on a number of independent random vectors whose distribution functions are not specified. It is proved that under certain regularity conditions, the regular and consistent estimates obtained are asymptotically normal as the number of random vectors tends to infinity. A necessary and sufficient condition for a regular and consistent estimate to have a "minimal" asymptotic covariance matrix is given. An expression is derived for the "minimal" asymptotic covariance matrix. It is also proved that if a function $\mathbf{f}$ satisfies certain conditions, then in order that $\mathbf{f}(\tilde\theta)$ be an RBAN estimate of $\mathbf{f}(\theta)$ at $\mathbf{f}(\theta^0)$, where $\theta^0$ is the true value of the parameter point $\theta$, it is necessary and sufficient that the argument $\tilde\theta$ be an RBAN estimate of $\theta$ at $\theta^0$. Methods of generating RBAN estimates are given. For simplicity of presentation, matrix notation is used throughout this paper. By derivatives of a matrix with respect to a vector (or with respect to a second matrix) is meant the derivatives of the matrix simultaneously with respect to all the components of the vector (or all the elements of the second matrix). The usual rules of differentiation with respect to vectors are used.