E-values for k-Sample Tests with Exponential Families

Sankhya A, vol. 86, pp. 596–636

Abstract

Abstract We develop and comparee-variables for testing whetherksamples of data are drawn from the same distribution, the alternative being that they come from different elements of an exponential family. We consider the GRO (growth-rate optimal)e-variables for (1) a ‘small’ null inside the same exponential family, and (2) a ‘large’ nonparametric null, as well as (3) ane-variable arrived at by conditioning on the sum of the sufficient statistics. (2) and (3) are efficiently computable, and extend ideas from Turner et al. (2021) and Wald (1947) respectively from Bernoulli to general exponential families. We provide theoretical and simulation-based comparisons of thesee-variables in terms of their logarithmic growth rate, and find that for small effects all foure-variables behave surprisingly similarly; for the Gaussian location and Poisson families, e-variables (1) and (3) coincide; for Bernoulli, (1) and (2) coincide; but in general, whether (2) or (3) grows faster under the alternative is family-dependent. We furthermore discuss algorithms for numerically approximating (1).

Authors 4

1. Yunda Hao

   Centrum Wiskunde & Informatica

   Affiliation as printed

   Centrum Wiskunde en Informatica, Amsterdam, The Netherlands

2. Peter Grünwald corresponding Aachen Mathematical Institute

   Leiden University · Centrum Wiskunde & Informatica

   Affiliation as printed

   Centrum Wiskunde en Informatica, Amsterdam, The Netherlands

   Mathematical Institute, Leiden University, Leiden, The Netherlands

3. Tyron Lardy Aachen Mathematical Institute

   Leiden University · Centrum Wiskunde & Informatica

   Affiliation as printed

   Centrum Wiskunde en Informatica, Amsterdam, The Netherlands

   Mathematical Institute, Leiden University, Leiden, The Netherlands

4. Long Long
   Affiliation as printed

   Yiyang, China