Robustness of estimation based on empirical transforms

The robustness of certain model-fitting procedures, based on statistical transforms, is investigated using the Influence Function. Our discussion is in two parts. In the first, we focus on estimateing the parameters of particular distributions, given independent and identically distributed realizations. We then move on, in the second part, to discuss the fitting of stochastic models.

In this latter context the approach baesd on transforms, such as the Laplace transform, offers the possibility of explicit parameter estimation. This is in obvious contrast to the more usual situation where only a numerical solution is possible. It was shown by Kemp & Kemp (1987), in a two-parameter example, that only a one-dimensional search was required to produce well-defined estimators. This phenomenon was noted earlier by Morgan (1982), and provides further motivation for transform methods. We generalize the result, and provide an example where a three-dimensional search can be reduced to a line search. With this in mind, we consider the fitting of stochastic models in some detail, employing the standard technique of ordinary least-squares as a bench-mark in this work.

The central theme of this thesis is, however, the robustness of such methods. To this end, we develop a powerful and flexible influence theory in the context of non-indexed random variables. These developments allow us to make concrete statements about the robustness of procedures based on transforms. We show that an analogous treatment is possible for the indexed case, allowing useful qualitative information about parameter estimators to be gathered.