Detecting the Presence of Mixing with Multiscale Maximum Likelihood

Abstract

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 re ects 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.

Global rates of convergence in log-concave density estimation

“…Log-concavity therefore offers statisticians the potential of freedom from restrictive parametric (typically Gaussian) assumptions without paying a hefty price. Indeed, in recent years, researchers have sought to exploit these alluring features to propose new methodology for a wide range of statistical problems, including the detection of the presence of mixing (Walther, 2002), tail index estimation (Müller and Rufibach, 2009), clustering (Cule, Samworth and Stewart, 2010), regression , Independent Component Analysis (Samworth and Yuan, 2012) and classification (Chen and Samworth, 2013).…”

“…For instance, the maximum likelihood estimator of a log-concave density, first studied by Walther (2002) in the case d = 1, and by Cule, Samworth and Stewart (2010) for general d, plays a central role in all of the procedures mentioned in the previous paragraph. Dümbgen, Hüsler and Rufibach (2011) developed a fast, Active Set algorithm for computing the estimator when d = 1, and this is implemented in the R package logcondens (Rufibach and Dümbgen, 2006;.…”

Estimating a Polya frequency function$_2$

“…Simulations are reported in Section 4. Rufibach and Duembgen [11] and Walther [14] adopt a similar approach but obtain different results by different methods.…”

Non-parametric log-concave mixtures

“…It also gives pleasing curves. Walther (2002) has a more general aim, taking only log-concavity as a goal, without considering smoothness. He also proposes a test for determining whether a data distribution might be a mixture.…”