Parameter estimation based on empirical transforms

In this thesis, we provide a unified treatment of the topic of parameter estimation using integral transforms, such as the characteristic function and moment generating function. This topic encompasses a wealth of methods, which typically vary from each other in relation to the type of weight function and choice of integral transform that is being employed.

We show that the integrated squared error method dominates alternative transform methods, particularly in terms of robustness. We present a convenient and

flexible approach to dealing with the difficulty here surrounding the necessary weight function, and illustrate the success of this approach on the mixture of two normal distributions. Furthermore, we show that the integrated squared error method also outperforms the maximum likelihood method for this distribution,

particularly with samples with outliers or a small number of observations.