Partial stochastic dominance for the multivariate Gaussian distribution

TL;DR

This paper proves a new partial stochastic dominance property for the maximum of multivariate Gaussian vectors with positive correlation, with applications in Bayesian clinical trial design.

Contribution

It establishes a novel partial stochastic dominance result for Gaussian maxima, extending the understanding of their distributional properties.

Findings

CDF of Gaussian maximum intersects standard normal CDF at most once

Result applies to Gaussian vectors with positive intraclass correlation

Useful for Bayesian clinical trial comparisons

Abstract

Gaussian comparison inequalities provide a way of bounding probabilities relating to multivariate Gaussian random vectors in terms of probabilities of random variables with simpler correlation structures. In this paper, we establish the partial stochastic dominance result that the cumulative distribution function of the maximum of a multivariate normal random vector, with positive intraclass correlation coefficient, intersects the cumulative distribution function of a standard normal random variable at most once. This result can be applied to the Bayesian design of a clinical trial in which several experimental treatments are compared to a single control.