Interval Spacing

Abstract

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.

Figures (5)

• Figure 1: Density of the interval spacing at widths $w$ of 2, 5, and 10 and indices $i$ as noted.
• Figure 2: Expected interval spacing for exponential and logistic variates at widths $w$ of 2, 5, and 10. Bands are inter-quartile ranges from simulations.
• Figure 3: Interval spacing is equivalent to a rectangular low-pass filter applied to spacing.
• Figure 4: Low-pass filtering of spacing for increasing interval widths $w$.
• Figure 5: Auto-covariance of the interval spacing follows the self-convolution of the rectangular kernel.