Consistency of Generalized Maximum Spacing Estimates

General methods for the estimation of distributions can be derived from approximations of certain information measures. For example, both the maximum likelihood (ML) method and the maximum spacing (MSP) method can be obtained from approximations of the Kullback–Leibler information. The ideas behind the MSP method, whereby an estimation method for continuous univariate distributions is obtained from an approximation based on spacings of an information measure, were used by Ranneby & Ekstrom (1997) (using simple spacings) and Ekstrom (1997b) (using high order spacings) to obtain a class of methods, called generalized maximum spacing (GMSP) methods. In the present paper, GMSP methods will be shown to give consistent estimates under general conditions, comparable to those of Bahadur (1971) for the ML method, and those of Shao & Hahn (1999) for the MSP method. In particular, it will be proved that GMSP methods give consistent estimates in any family of distributions with unimodal densities, without any further conditions on the distributions.

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