Bivariate Analysis of Distribution Functions Under Biased Sampling

Abstract

This article compares distribution functions among pairs of locations in their domains, in contrast to the typical approach of univariate comparison across individual locations. This bivariate approach is studied in the presence of sampling bias, which has been gaining attention in COVID-19 studies that over-represent more symptomatic people. In cases with either known or unknown sampling bias, we introduce Anderson–Darling-type tests based on both the univariate and bivariate formulation. A simulation study shows the superior performance of the bivariate approach over the univariate one. We illustrate the proposed methods using real data on the distribution of the number of symptoms suggestive of COVID-19.

KEYWORDS:

• Bootstrap
• Empirical distribution function
• Size bias
• Two-sample test

Supplementary Materials

We provide technical details on the AD-type tests, additional figures and numerical results, and extensions to other statistics in the Supplement. R code for implementing all the methods and for reproducing the results in Sections 3 and 4 can be obtained online from the journal’s website.

Acknowledgments

The authors thank Yi-Ching Yao for inspiring the study of the bivariate formulation, Da-Wei Huang for computational support, Shih-Hao Huang for helpful comments, and Wordvice for language proofreading and editing service.

Disclosure Statement

The authors report there are no competing interests to declare.

Additional information

Funding

The research of Hsin-wen Chang was partially supported by National Science and Technology Council of Taiwan under Grant 110-2118-M-001-013-MY2.