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<p>In environmental studies, one may be interested in examining whether cities with similar air quality index (AQI) values also have comparable air pollutant concentrations. To understand the intricate relationships between two species, ecologists need to measure the degree of overlap between two multivariate distributions representing their ecological niches. The requirements of multivariate normality and/or the equality of covariance matrices are limiting the usability of current inferential methods for the study of similarity (overlap) between two multivariate distributions. In this work, we extend newly developed containment and overlap indices for the multivariate case and propose new measures to quantify the containment of a multivariate distribution within another, as well as the overlap between them. Point and interval estimation procedures are developed for these measures for families of elliptical distributions. Consistency of the proposed estimators is proved under some mild conditions. Furthermore, the bias and risk performance of these estimators with respect to two different loss functions are investigated numerically. Bootstrap confidence intervals of these measures are also given, and a simulation study is performed to see their confidence bands under different parametric configurations. An open source ‘<Emphasis FontCategory="NonProportional">R</Emphasis>’ software package has been created for practitioners to obtain point and interval estimates of the proposed measures for their data sets. An illustration using AQI data demonstrates how to apply the method in order to analyze overlap of air quality between different cities.</p>

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A new multivariate overlap index for elliptical distributions: analyzing air quality in cities

  • 期刊论文

  • 发表日期：

    2026年5月5日

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  • Raju Dey,
  • Arne C. Bathke,
  • Somesh Kumar

摘要

In environmental studies, one may be interested in examining whether cities with similar air quality index (AQI) values also have comparable air pollutant concentrations. To understand the intricate relationships between two species, ecologists need to measure the degree of overlap between two multivariate distributions representing their ecological niches. The requirements of multivariate normality and/or the equality of covariance matrices are limiting the usability of current inferential methods for the study of similarity (overlap) between two multivariate distributions. In this work, we extend newly developed containment and overlap indices for the multivariate case and propose new measures to quantify the containment of a multivariate distribution within another, as well as the overlap between them. Point and interval estimation procedures are developed for these measures for families of elliptical distributions. Consistency of the proposed estimators is proved under some mild conditions. Furthermore, the bias and risk performance of these estimators with respect to two different loss functions are investigated numerically. Bootstrap confidence intervals of these measures are also given, and a simulation study is performed to see their confidence bands under different parametric configurations. An open source ‘R’ software package has been created for practitioners to obtain point and interval estimates of the proposed measures for their data sets. An illustration using AQI data demonstrates how to apply the method in order to analyze overlap of air quality between different cities.