Detecting the presence of mixing with multiscale maximum likelihood

A test of homogeneity tries to decide whether observations come from a single distribution or from a mixture of several distributions. A powerful theory has been developed for the case where the component distributions are members of an exponential family. When no parametric assumptions are appropriate, the standard approach is to test for bimodality, which is known not to be very sensitive for detecting heterogeneity. To develop a more sensitive procedure, this article builds on an approach employed in sampling literature and models the component distributions as logarithmically concave densities. It is shown how this leads to a special semiparametric model, in which the homogeneity problem is equivalent to testing whether a parameter c equals zero, versus the alternative that c > 0. This setup leads naturally to a novel multiscale maximum likelihood procedure, where the multiscale character reflects the desirable property of adaptivity to the unknown value of the parameter c under the alternative, to ensure a test with high power. The test can also be extended, in principle, to a multivariate situation. In a univariate setting, the multiscale procedure is well suited to computation with the iterative convex minorant algorithm, a recent innovation in computational nonparametric statistics. The test is applied to simulated data and to the stamp data of Izenman and Sommer.