The Maximum Spacing Method. An Estimation Method Related to the Maximum Likelihood Method

The maximum likelihood method (ML method) works properly if each contri- bution to the likelihood function is bounded from above. This is the case for all discrete distributions but not for all mixtures of continuous distributions and then the ML method can break down. Since the ML method can be obtained by approximating the Kullback-Leibler information we raise the following question. Is it possible to obtain better methods than the ML method by approximating the Kullback-Leibler information in another way? Using spacings we can obtain an approximation of the Kullback-Leibler information such that each component in the approximation is bounded from above. In this paper we present the method, which we call the maximum spacing method, and show consistency of the maximum spacing estimate. Some examples are given, which show that the new method works also in situations where the ML method breaks down.