distributionranging

Distributionranging is a statistical concept that analyzes how the distribution of a variable changes when observations are restricted to different value ranges. It is used to explore tail behavior, detect subpopulations, and assess sensitivity to the choice of range.

Formally, for a random variable X with distribution F, distributionranging examines the family of truncated distributions X conditioned on a ≤ X ≤ b, parameterized by the interval [a, b]. For each interval, one can compute statistics such as the mean, variance, skewness, and higher moments, or estimate the conditional density f_{a,b}(x). By sweeping a and b over a predefined grid, researchers obtain curves of statistics as functions of range length or interval location, revealing how the distribution shifts with range.

Common approaches include fixed-length windows that slide across the support of X and variable-length windows with a chosen center. Interpreting the results involves assessing tail heaviness, multimodality, and the presence of heterogeneous subpopulations, as well as enabling comparisons between groups by aligning their ranges.

Applications span finance (analyzing return or loss distributions within thresholds), environmental and quality-control data (range-limited measurements), risk assessment (stress-testing over ranges), and survey analysis (responses within specified scales). Distributionranging relates to truncation and censoring, order statistics, and robust statistics, offering a framework for examining distributional features that are not visible when considering the full range alone.

Limitations include potential bias from arbitrary range choices, reduced data in narrow intervals, and increased computational demands. See also truncation, censoring, and order statistics.