2025/09/15 by Chakraborty, Parijat, Guo, F. Richard, Shedden, Kerby +1
#Applications (stat.AP) #FOS: Computer and information sciences #FOS: Mathematics #Methodology (stat.ME) #Probability (math.PR) #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.2509.12066
It is often of interest to test a global null hypothesis using multiple, possibly dependent, p-values by combining their strengths while controlling the Type I error. Recently, several heavy-tailed combinations tests, such as the harmonic mean test and the Cauchy combination test, have been proposed: they map p-values into heavy-tailed random variables before combining them in some fashion into a single test statistic. The resulting tests, which are calibrated under the assumption of independence of the p-values, have shown to be rather robust to dependence. The complete understanding of the calibration properties of the resulting combination tests of dependent and possibly tail-dependent p-values has remained an important open problem in the area. In this work, we show that the powerful framework of multivariate regular variation (MRV) offers a nearly complete solution to this problem. We first show that the precise asymptotic calibration properties of a large class of homogeneous combination tests can be expressed in terms of the angular measure -- a characteristic of the asymptotic tail-dependence under MRV. Consequently, we show that under MRV, the Pareto-type linear combination tests, which are equivalent to the harmonic mean test, are universally calibrated regardless of the tail-dependence structure of the underlying p-values. In contrast, the popular Cauchy combination test is shown to be universally honest but often conservative; the Tippet combination test, while being honest, is calibrated if and only if the underlying p-values are tail-independent. One of our major findings is that the Pareto-type linear combination tests are the only universally calibrated ones among the large family of possibly non-linear homogeneous heavy-tailed combination tests.