Maximum Spacing Estimation - History and Usage

History and Usage

The MSE method was derived independently by Russel Cheng and Nik Amin at the University of Wales Institute of Science and Technology, and Bo Ranneby at the Swedish University of Agricultural Sciences. The authors explained that due to the probability integral transform at the true parameter, the “spacing” between each observation should be uniformly distributed. This would imply that the difference between the values of the cumulative distribution function at consecutive observations should be equal. This is the case that maximizes the geometric mean of such spacings, so solving for the parameters that maximize the geometric mean would achieve the “best” fit as defined this way. Ranneby (1984) justified the method by demonstrating that it is an estimator of the Kullback–Leibler divergence, similar to maximum likelihood estimation, but with more robust properties for various classes of problems.

There are certain distributions, especially those with three or more parameters, whose likelihoods may become infinite along certain paths in the parameter space. Using maximum likelihood to estimate these parameters often breaks down, with one parameter tending to the specific value that causes the likelihood to be infinite, rendering the other parameters inconsistent. The method of maximum spacings, however, being dependent on the difference between points on the cumulative distribution function and not individual likelihood points, does not have this issue, and will return valid results over a much wider array of distributions.

The distributions that tend to have likelihood issues are often those used to model physical phenomena. Hall & al. (2004) seek to analyze flood alleviation methods, which requires accurate models of river flood effects. The distributions that better model these effects are all three-parameter models, which suffer from the infinite likelihood issue described above, leading to Hall’s investigation of the maximum spacing procedure. Wong & Li (2006), when comparing the method to maximum likelihood, use various data sets ranging from a set on the oldest ages at death in Sweden between 1905 and 1958 to a set containing annual maximum wind speeds.