Estimation with samples drawn from different but parametrically related distributions

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Virginia Polytechnic Institute


This dissertation discusses a method of estimation with random samples drawn from two different normal populations. The two populations may be either univariate or multivariate (provided the populations have the same number of variates) and they are to be parametrically related in that the means (or vector of means) are equal. Since it is assumed that the two normal populations have a common mean (or a common vector of means), these common parameters are jointly estimated along with all the other unknown parameters. The joint estimates, called iteration estimates, are obtained by an iteration method developed for solving the likelihood equations.

A detailed study is made for the joint estimation of parameters when sampling from two univariate normal populations. The iteration procedure is based on jointly estimating the common mean and the individual variances by finding a weighted mean and the individual variances about the weighted mean. The initial weighted mean is found by taking as weights the reciprocals of the estimates of the variances of the individual estimates of the mean. It is proved that the iteration method produces a unique set of estimates which satisfies the likelihood equations. Since this set of estimates is not always identical with the set of maximum likelihood estimates, the conditions under which the two sets may possibly differ are established. Numerical examples are given to illustrate the iteration technique and to compare the iteration estimates with maximum likelihood estimates in the cases where they differ.

Empirical sampling, with small sizes, is done with the aid of the IBM 650 Computer to obtain information regarding the distribution of the iteration estimates and also the maximum likelihood estimates in the cases where they differ. The experimental results indicate that the iteration estimate of the common mean tends to be normally distributed and the iteration estimates of the individual variances are virtually unbiased.

The iteration procedure is compared with Fisher’s Method, which uses the Information Matrix, and is shown to give identical results while requiring less computation.

An extension of the iteration procedure is made to the case where the samples are drawn from two bivariate normal populations with the components of the common vector of means and the elements of the individual covariance matrices being estimated jointly. For the particular case in which the individual variances within each population may be assumed equal, it is shown that a linear transformation to obtain new uncorrelated variables will materially lessen the time required for the iteration method. A numerical example is given to illustrate the iteration technique both with and without a transformation of variables and a proof is given to show that the two methods produce identical results.

The iteration procedure is further extended to the case where the samples are drawn from two multivariate normal populations which have the same number of variates and joint estimates are obtained for the common vector of means and the individual covariance matrices. It is also shown that if a linear transformation can be found which gives new uncorrelated variables in each population, then transformation before iteration greatly reduces the computational labor involved in obtaining the joint estimates.