Groupings in item demand problems

In this dissertation an iterative procedure, due to Hartley [9], for obtaining the maximum likelihood estimators of the parameters from underlying discrete distributions is studied for the case of grouped random samples. It is shown that when the underlying distribution is Poisson the process always converges and does so regardless of the initial values taken for the unknown parameter. In showing this, a rather interesting property of the Poisson distribution was derived. If one defines a connected group of integers to be such that it contains all the integers between and including its end points, it is shown that the variance of the sub- distribution defined on this connected set is strictly less than the variance of the complete Poisson distribution. A Monte Carlo study was performed to indicate how increasing group sizes affected the variances of the maximum likelihood estimators.

As a result of a problem encountered by the Office of Naval Research, combinations of distributions diff kb were introduced. The difference between such combinations and the classical mixtures of distributions is that a new distribution must be considered whenever the random variable in question increases by an integral multiple of a known integer constant, b. When all the data are present, the estimation problem is no more complicated than when estimating the individual parameters from the component distributions. However, it is pointed out that very frequently the observed samples are defective in the fact that none of the component frequencies are observed. Hence, horizontal grouping of the sample values occurs as opposed to the vertical grouping encountered previously in the one parameter Poisson case. An extension of the iterative procedure used to obtain the maximum likelihood estimator of the single parameter grouped Poisson distribution is made to obtain the estimators of the parameters in a horizontally grouped sample.

As a practical example, the component distributions were all taken to be from the Poisson family. The estimators were obtained and their properties were studied. The regularity conditions which are sufficient to show that a consistent and asymptotically normally distributed solution to the likelihood equations exist are seen to be satisfied for such combinations of the Poisson distributions. Further, in the full data case, a set of jointly sufficient statistics is exhibited and since, in the presence of sufficient statistics, the solutions to the likelihood equations are unique, the estimators are consistent and asymptotically normal.

It is seen that such combinations of distributions can be applied to problems in item demands. A justification of the Poisson distribution is given for such applications, but it is also pointed out that the Negative Binomial distribution might be applicable. It is also shown that such a probability model might have an application in testing the efficiency of an anti-ballistic missile system when under attack by missiles which carry multiple warheads. However, no data were available and hence the study of this application could be carried no further.