In statistics, maximum spacing estimation (MSE or MSP), or maximum product of spacing estimation (MPS), is a method for estimating the parameters of a univariate statistical model. The method requires maximization of the geometric mean of spacings in the data, which are the differences between the values of the cumulative distribution function at neighbouring data points. The concept underlying the method is based on the probability integral transform, in that a set of independent random samples derived from any random variable should on average be uniformly distributed with respect to the cumulative distribution function of the random variable. The MPS method chooses the parameter values that make the observed data as uniform as possible, according to a specific quantitative measure of uniformity. One of the most common methods for estimating the parameters of a distribution from data, the method of maximum likelihood (MLE), can break down in various cases, such as involving certain mixtures of continuous distributions. In these cases, the method of maximum spacing estimation may be successful. Apart from its use in pure mathematics and statistics, the trial applications of the method have been reported using data from fields such as hydrology, econometrics, magnetic resonance imaging, and others.
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 some 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 require 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, uses 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.
Definition Given an Independent and identically random sample { x 1 , … , x n } {\displaystyle \{x_{1},\dots ,x_{n}\}} of size n {\displaystyle n} from a univariate distribution with continuous cumulative distribution function F ( x ; θ 0 ) {\displaystyle F(x;\theta _{0})} , where θ 0 ∈ Θ {\displaystyle \theta _{0}\in \Theta } is an unknown parameter to be estimated, let { x ( 1 ) , … , x ( n ) } {\displaystyle \{x_{(1)},\dots ,x_{(n)}\}} be the corresponding ordered sample, that is the result of sorting of all observations from smallest to largest. Denote x ( 0 ) = inf { S } {\displaystyle x_{(0)}=\inf\{S\}} and x ( n + 1 ) = sup { S } {\displaystyle x_{(n+1)}=\sup\{S\}} , where S {\displaystyle S} denotes the support of the distribution. Define the spacings as the “gaps” between the values of the distribution function at adjacent ordered points:
D i ( θ ) = F ( x ( i ) ; θ ) − F ( x ( i − 1 ) ; θ ) , i = 1 , … , n + 1. {\displaystyle D_{i}(\theta )=F(x_{(i)};\,\theta )-F(x_{(i-1)};\,\theta ),\quad i=1,\ldots ,n+1.}
Then the maximum spacing estimator of θ 0 {\displaystyle \theta _{0}} is defined as a value that maximizes the logarithm of the geometric mean of sample spacings:
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