## Doctoral Dissertations

#### Title

Maximum likelihood estimation of random sample size models

January 1999

Statistics

Ph.D.

#### Abstract

Sometimes in an experiment, a random number of sample events are observed as opposed to the usual fixed number of events. The random sample size can contain useful experimental information which can be used to estimate the parameters of the desired model. ^ To demonstrate the concepts, consider an example from the study of electrical insulation. A segment of insulation is subjected to a voltage stress which causes cracks in the insulation. The number of cracks as well as the lengths of the cracks is then recorded. ^ This experiment can be repeated over an independent variable, such as separate voltages. and a mod developed. A traditional approach could be to regress the number of cracks, m, on voltage, v, Em&vbm0;v=b0+b1 v and then regress the lengths of the cracks, y, versus, v, similarly Ey&vbm0;v=b ′′0+b′′ 1v Estimating the change in “crack damage”, with respect to v, should involve estimating b0, and b′′1. ^ The approach I wish to discuss is to regress the two sets of values together. This approach is more reasonable than the previously discussed approach when the relationship between the means of the populations of the Random Sample Size and Sample Event is proportional with respect to the independent variable. A possible model Em&vbm0;v=b 0+b1v and Ey&vbm0;v=a b0+b1v with α the ratio of the two means. ^ The new model is more desirable because the change in “crack damage” is principally modeled by β1. The estimate for this parameter will involve both the number of cracks and the lengths of the cracks and hence use more information to estimate the desired parameter. ^ In this work, we will also discuss more general models. I intend to investigate several models and demonstrate under what conditions the MLE exists and is a unique critical point. ^ This thesis will take a standard approach to maximum likelihood estimation. First, we will develop asymptotic properties of MLE's for our types of models. Finally, the existence and uniqueness of maximums of the likelihood of specific models will be discussed. ^

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