Maximum Spacing Estimation - Definition

Definition

Given an iid random sample {x1, …, xn} of size n from a univariate distribution with the cumulative distribution function F(x;θ0), where θ0 ∈ Θ is an unknown parameter to be estimated, let {x(1), …, x(n)} be the corresponding ordered sample, that is the result of sorting of all observations from smallest to largest. For convenience also denote x(0) = −∞ and x(n+1) = +∞.

Define the spacings as the “gaps” between the values of the distribution function at adjacent ordered points:

D_i(\theta) = F(x_{(i)};\,\theta) - F(x_{(i-1)};\,\theta), \quad i=1,\ldots,n+1.

Then the maximum spacing estimator of θ0 is defined as a value that maximizes the logarithm of the geometric mean of sample spacings:

\hat{\theta} = \underset{\theta\in\Theta}{\operatorname{arg\,max}} \; S_n(\theta), \quad\text{where }\ S_n(\theta) = \ln\!\! \sqrt{D_1D_2\cdots D_{n+1}} = \frac{1}{n+1}\sum_{i=1}^{n+1}\ln{D_i}(\theta).

By the inequality of arithmetic and geometric means, function Sn(θ) is bounded from above by −ln(n+1), and thus the maximum has to exist at least in the supremum sense.

Note that some authors define the function Sn(θ) somewhat differently. In particular, Ranneby (1984) multiplies each Di by a factor of (n+1), whereas Cheng & Stephens (1989) omit the 1⁄n+1 factor in front of the sum and add the “−” sign in order to turn the maximization into minimization. As these are constants with respect to θ, the modifications do not alter the location of the maximum of the function Sn.

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