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A Spacing Estimator

Greg Kreider

TL;DR

The paper extends spacing analyses of order statistics beyond uniform and exponential to include logistic and Gumbel variates, and proposes a practical quantile-based estimator for the expected spacing when the inverse CDF is known. It shows that the estimator is exact for exponential and logistic distributions, with a concise closed-form for the logistic case, while Gumbel requires approximate treatment; the method provides accurate mid-distribution spacing (high accuracy for moderate to large $n$) but degrades in the tails, particularly for asymmetric variates. Analytic results are provided for uniform, exponential, logistic, and Gumbel spacings, with extensive Appendices detailing derivations and the relationship between the estimator and the exact spacing moments. The work offers a usable approach to estimate spacings in a variety of distributions, with clear guidance on expected accuracy and limitations in tail regions, beneficial for statistical inference and related applications.

Abstract

The distribution of the spacing, or the difference between consecutive order statistics, is known only for uniform and exponential random variates. We add here logistic and Gumbel variates, and present an estimator for distributions with a known inverse cumulative density function. We show the estimator is accurate to the limit of numerical simulations for points near the middle of the order statistics, but degrades by up to 20% in the tails.

A Spacing Estimator

TL;DR

The paper extends spacing analyses of order statistics beyond uniform and exponential to include logistic and Gumbel variates, and proposes a practical quantile-based estimator for the expected spacing when the inverse CDF is known. It shows that the estimator is exact for exponential and logistic distributions, with a concise closed-form for the logistic case, while Gumbel requires approximate treatment; the method provides accurate mid-distribution spacing (high accuracy for moderate to large ) but degrades in the tails, particularly for asymmetric variates. Analytic results are provided for uniform, exponential, logistic, and Gumbel spacings, with extensive Appendices detailing derivations and the relationship between the estimator and the exact spacing moments. The work offers a usable approach to estimate spacings in a variety of distributions, with clear guidance on expected accuracy and limitations in tail regions, beneficial for statistical inference and related applications.

Abstract

The distribution of the spacing, or the difference between consecutive order statistics, is known only for uniform and exponential random variates. We add here logistic and Gumbel variates, and present an estimator for distributions with a known inverse cumulative density function. We show the estimator is accurate to the limit of numerical simulations for points near the middle of the order statistics, but degrades by up to 20% in the tails.
Paper Structure (8 sections, 157 equations, 3 figures, 1 table)

This paper contains 8 sections, 157 equations, 3 figures, 1 table.

Table of Contents

  1. Spacings
  2. Spacing Estimator
  3. Conclusion
  4. Spacing for Uniform Variates
  5. Spacing for Exponential Variates
  6. Spacing for Logistic Variates
  7. Spacing for Gumbel Variates
  8. Matching of Logistic Expected Spacing to Quantile Estimator

Figures (3)

  • Figure 1: Difference between the mean spacing and quantile estimator for the $p$ distributions: the exponential, logistic, Laplace, and Pareto.
  • Figure 2: Difference between the mean spacing and quantile estimator for the $p$, $\ln p$ distributions: the Gumbel, Rayleigh, Weibull, and Frechet.
  • Figure 3: Difference between the mean spacing and quantile estimator for the other distribution: the Cauchy.