Assessing the similarity of distributions - finite sample performance of the empirical mallows distance

Abstract

The problem of assessing similarity of two cumulative distribution functions (c.d.f.'s) has been the topic of a previous paper by the authors (Munk & Czado (1995)). Here, we developed an asymptotic test based on a trimmed version of the Mallows distance (Mallows 1972) between two c.d.f.'s F and G. This allows to assess the similarity of two c.d.f.'s with respect to this distance at controlled type I error rate. In particular, this applies to bioequivalence testing within a purely nonparametric setting. In this paper, we investigate the nite sample behavior of this test. The e ect of trimming and non equal sample size on the observed power and level is studied. Sample size driven recommendations for the choice of the trimming bounds are given in order to minimize the bias. Finally, assuming normality and homogeneous variances, we simulate the relative e ciency of the Mallows test to the (asymptotically optimal) standard equivalence t test, which reveals the Mallows test as a robust alternative to the standard equivalence t test.

Nonparametric Validation of Similar Distributions and Assessment of Goodness of Fit

“…and X 0 X X 1 and Y 0 X Y 1 . To investigate the small sample properties of the Mallows equivalence test we performed detailed Monte Carlo studies which are presented comprehensively in Czado and Munk (1996). We brie¯y summarize the basic results which address the following questions.…”

“…A key issue in the nonparametric bioequivalence discussion is the sample size required to control a small probability of type II error. Therefore Czado and Munk (1996) simulated the relative eciency of the Mallows equivalence test and the standard equivalence test under a linear model with additive normal errors. Typically, such a model occurs after a logarithmic transformation of the raw data (such as the area under the blood concentration±time curve or the time to achieve maximal concentration) where the ratio of the means must be within some reasonable limits.…”

Central limit theorem and bootstrap procedure for Wasserstein’s variations with an application to structural relationships between distributions

“…Under some mild conditions, a deformation model holds if and only if this minimal alignment cost is null and we can base our assessment of a deformation model on this quantity. As in [15,25], we provide results (a Central Limit Theorem and bootstrap versions) that enable to reject that the minimal alignment cost exceeds some threshold, and hence to conclude that it is below that threshold. Our results are given in a setup of general, nonparametric classes of warping functions.…”

“…We note that within this framework, statistical inference on deformation models for distributions has been studied first in [20]. Here we provide a different approach which allows to deal with more general deformation classes.The pioneering works [15,25] study the existence of relationships between distributions F and G by using a discrepancy measure ∆(F, G) between them which is built using the Wasserstein distance. The authors consider the assumption H 0 : ∆(F, G) > ∆ 0 versus H a : ∆(F, G) ≤ ∆ 0 for a chosen threshold ∆ 0 .…”

A multivariate test for population bioequivalence

“…This is reasonable in the context of evaluation of generic pharmaceutical products where the reference is the current product, but is a potential limitation when moving outside of this area. Examples of symmetric criteria for the univariate and multivariate cases are found in References [28,29].…”