Smooth tests of hypothesis for problems in high dimension

Abstract

Hypothesis tests are largely used by the scientific community. In particular, goodness of fit tests are extremely important, because they allow us to test if scientific theories adequately describe the framework of a dataset. Recently, several tests for high-dimensional data have been proposed, which enables these statistical methodology to be applied in complex objects such as images, trajectories and SNPs. Yet, these test aren't easily interpreted. In this work, we overcome this problem by proposing a generalization of the Neyman's soft test, originally designed for univariate problems. This scenario shows a challenge for the traditional approach, since the Neyman's test is based on an expansion of the data density on an orthonormal basis, and usual bases are difficult to treat in high-dimensional problems. A solution to overmatch this problem is to use spectral bases to model the multivariate distribution of the data. From it, we use the ideas behind the Neyman's test to test high-order goodness of fit hypotheses. We will also make several simulation experiments comparing our proposal with other tests in the literature.