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.

Read more about this topic:  Maximum Spacing Estimation

Famous quotes containing the word definition:

    “It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
    —Jane Adams (20th century)

    “It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
    —François, Duc De La Rochefoucauld (1613–1680)

    “Scientific method is the way to truth, but it affords, even in
    principle, no unique definition of truth. Any so-called pragmatic
    definition of truth is doomed to failure equally.”
    —Willard Van Orman Quine (b. 1908)