Andrea Ongaro

Adjusting p-Values when n Is Large in the Presence of Nuisance Parameters

A precise null hypothesis formulation (instead of the more realistic interval one) is usually adopted by statistical packages although it generally leads to excessive (and often misleading) rates of rejection whenever the sample size is large. In a previous paper (Migliorati and Ongaro, 2007) we proposed a calibration procedure aimed at adjusting test levels and p-values when testing the mean of a Normal model with known variance. We now address the more complicated calibration issues arising when a nuisance parameter (e.g., the variance) is present. As procedures for testing the interval null hypothesis available in the literature are shown to be unsatisfactory for calibration purposes, this entails, in particular, the construction of suitable new tests.

A generalization of the Dirichlet distribution

A new parametric family of distributions on the unit simplex is proposed and investigated. Such family, called flexible Dirichlet, is obtained by normalizing a correlated basis formed by a mixture of independent gamma random variables. The Dirichlet distribution is included as an inner point. The flexible Dirichlet is shown to exhibit a rich dependence pattern, capable of discriminating among many of the independence concepts relevant for compositional data. At the same time it can model multi-modality. A number of stochastic representations are given, disclosing its remarkable tractability. In particular, it is closed under marginalization, conditioning, subcomposition, amalgamation and permutation.

A structured Dirichlet mixture model for compositional data: inferential and applicative issues

The flexible Dirichlet (FD) distribution (Ongaro and Migliorati in J. Multvar. Anal. 114: 412–426, 2013) makes it possible to preserve many theoretical properties of the Dirichlet one, without inheriting its lack of flexibility in modeling the various independence concepts appropriate for compositional data, i.e. data representing vectors of proportions. In this paper we tackle the potential of the FD from an inferential and applicative viewpoint. In this regard, the key feature appears to be the special structure defining its Dirichlet mixture representation. This structure determines a simple and clearly interpretable differentiation among mixture components which can capture the main features of a large variety of data sets. Furthermore, it allows a substantially greater flexibility than the Dirichlet, including both unimodality and a varying number of modes. Very importantly, this increased flexibility is obtained without sharing many of the inferential difficulties typical of general mixtures. Indeed, the FD displays the identifiability and likelihood behavior proper to common (non-mixture) models. Moreover, thanks to a novel non random initialization based on the special FD mixture structure, an efficient and sound estimation procedure can be devised which suitably combines EM-types algorithms. Reliable complete-data likelihood-based estimators for standard errors can be provided as well.

A Dirichlet Mixture Model for Compositions Allowing for Dependence on the Size

The Dirichlet is the most well known distribution for compositional data, i.e. data representing vectors of proportions. The flexible Dirichlet distribution (FD) generalizes the Dirichlet one allowing to preserve its main mathematical and compositional properties. At the same time, it does not inherit its lack of flexibility in modeling the dependence concepts appropriate for compositional data. The present paper introduces a new model obtained by extending the basis of positive random variables generating the FD by normalization. Specifically, the new basis exhibits a more sophisticated mixture (latent) representation, which leads to a twofold result. On the one side, a more general distribution for compositional data, called EFD, is obtained by normalization. In particular, the EFD allows for a significantly wider differentiation among the clusters defining its mixture representation. On the other side, the generalized basis induces a tractable model for the dependence between composition and size: the conditional distribution of the composition given the size is still an EFD, the size affecting it in a simple fashion through the cluster weights.

On the topological support of species sampling priors

Abstract: In Bayesian nonparametric statistics, it is crucial that the support of the prior is very large. Here, we consider species sampling priors. Such priors are widely used within mixture models and it has been shown in the literature that a large support for the mixing prior is essential to ensure the consistency of the posterior. In this paper, simple conditions are given that are necessary and sufficient for the support of a species sampling prior to be full. In particular, for proper species sampling priors, the condition is that the maximum size of the atoms of the corresponding process is small with positive probability. We apply this result to show that the main classes of species sampling priors known in literature have full support under mild conditions. Moreover, we find priors with a very simple construction still having full support.