Alternatives to maximum likelihood estimation based on spacings and the Kullback–Leibler divergence

An alternative to the maximum likelihood (ML) method, the maximum spacing (MSP) method, is introduced in Cheng and Amin [1983. Estimating parameters in continuous univariate distributions with a shifted origin. J. Roy. Statist. Soc. Ser. B 45, 394–403], and independently in Ranneby [1984. The maximum spacing method. An estimation method related to the maximum likelihood method. Scand. J. Statist. 11, 93–112]. The method, as described by Ranneby [1984. The maximum spacing method. An estimation method related to the maximum likelihood method. Scand. J. Statist. 11, 93–112], is derived from an approximation of the Kullback–Leibler divergence. Since the introduction of the MSP method, several closely related methods have been suggested. This article is a survey of such methods based on spacings and the Kullback–Leibler divergence. These estimation methods possess good properties and they work in situations where the ML method does not. Important issues such as the handling of ties and incomplete data are discussed, and it is argued that by using Moran's [1951. The random division of an interval—Part II. J. Roy. Statist. Soc. Ser. B 13, 147–150] statistic, on which the MSP method is based, we can effectively combine: (a) a test on whether an assigned model of distribution functions is correct or not, (b) an asymptotically efficient estimation of an unknown parameter θ0θ0, and (c) a computation of a confidence region for θ0θ0.