PIMS MSS Colloquium: Daniel Hlubinka

Observing a sample of random variables, we wish to decide whether or not some hypothesis about the distribution of the observed sample is correct. A decision procedure for such a problem is called a statistical test of the hypothesis under consideration. We focus on nonparametric tests, the construction of which requires only minimal assumptions on the underlying probability distribution.

Linear rank test statistics form a large family of tests for hypotheses in linear models including the well-known two-sample location comparison, ANOVA, and linear regression. To calculate a rank statistics for one-dimensional random variables one needs to order the data with respect to its value or absolute value. A natural linear ordering is available only for one-dimensional data, while for multivariate data there are several definitions of linear ordering, however, no one is a direct extension of the linear ordering on the real line. Thus, extending rank-based inference to a multivariate setting such as multiple-output regression or MANOVA with an unspecified d-dimensional error density has remained an open problem for more than half a century.

A concept of center-outward multivariate ranks and signs based on measure transportation ideas has been introduced recently. Center-outward ranks and signs are not only distribution-free, but in dimension d > 1 achieve the (essential) maximal ancillarity property of traditional univariate ranks. In the present case, we show that fully distribution-free testing procedures based on center-outward ranks can achieve parametric efficiency. We establish the H´ajek representation and asymptotic normality results required in the construction of such tests in multiple-output regression and MANOVA models. Simulations and an empirical study demonstrate the excellent performance of the proposed procedures.