An Application of Linearization in Nonparametric Multivariate Analysis

Attention is restricted to two-dimensional random vectors. The underlying bivariate random structure can be investigated by means of statistics based on the linearized sample elements; the linear compounds are parametrized by the angle that the compounding unit vector makes with the positive first coordinate axis. The collection of statistics forms a stochastic process, certain functionals of which will be relevant statistics for the problem in its original bivariate setting. In this paper we focus on an application to the nonparametric bivariate regression problem, which is very similar to the union-intersection method employed in Roy (1953) and Roy and Bose (1953) in a parametric setting. In the present case the statistics are simple linear rank statistics based on the ranks of the linearized sample elements. The asymptotic distribution theory is developed under the null hypothesis and is to a large extent almost immediate from results in Hájek and Šidàk (1967).