Assessing structural relationships between distributions - a quantile process approach based on Mallows distance

We suggest diagnostic tools for the assessment of shape similarity of distributions, i.e. whether several distributions are equal up to a pre-specified semiparametric deviation, such as a simple shift in location or a scale deviation. This will be called a structural relationship between distributions, and it is of importance, for instance, in an initial analysis within the (nonparametric) analysis of variance. Our approach is based on a modification of a trimmed version of Mallows distance between distribution functions. Asymptotic theory for the corresponding test statistics is provided and a bootstrap limit theorem is shown. In particular, the case of dependent samples is covered as well. A simulation study reveals the bias corrected and accelerated bootstrap as an adequate method for the assessment of shape similar distributions. A medical application is discussed.