Interval Spacing

We define interval spacing as the difference in the order statistics of data over a gap of some width. We derive its density, expected value, and variance for uniform, exponential, and logistic variates. We show that interval spacing is equivalent to running a rectangular low-pass filter over the spacing, which simplifies the expressions for the expected values and introduces correlations between overlapping intervals.

💡 Research Summary

The manuscript introduces “interval spacing,” a generalization of the classic spacing between consecutive order statistics, defined as the difference between the i‑th order statistic T_i and the order statistic w positions earlier, D_{i,w}=T_i−T_{i−w}. While the case w=1 (ordinary spacing) is well studied, the authors develop a full analytical framework for arbitrary positive integer widths w. Starting from the joint density of two order statistics (Wilks, 1948), they derive a compact expression for the density f_{D_{i,w}}(y) (Equation 2) that involves the underlying distribution function F(x) raised to the power w−1. This formulation naturally reduces to the known spacing density when w=1.

The paper then specializes the general result to three canonical families: uniform, exponential, and logistic distributions.

• Uniform case: Closed‑form expressions are obtained for the density (Equation 5), the mean E