Pietro Muliere

A Note on Stochastic Dominance and Inequality Measures

A sequence of partial orders (called inverse stochastic dominances) is introduced on the set of distribution functions (of nonnegative random variables). The partial orders previously defined are used to rank income distributions when Lorenz ordering does not hold, i.e., when Lorenz curves intersect. It is known that the Gini index is coherent with second degree stochastic dominance (and with second degree inverse stochastic dominance). It will be shown that it is coherent with third degree inverse stochastic dominance, too. It will finally be shown that a sequence of ethically flexible Gini indices due to D. Donaldson and J. A. Weymark (Ethically flexible Gini indices for income distribution in the continuum, J. Econ. Theory 29 (1983), 353–356) is coherent with the sequence of nth degree inverse stochastic dominances

Approximating distributions of random functionals of Ferguson‐Dirichlet priors

We explore the possibility of approximating the Ferguson-Dirichlet prior and the distributions of its random functionals through the simulation of random probability measures. The proposed procedure is based on the constructive definition illustrated in Sethuraman (1994) in conjunction with the use of a random stopping rule. This allows us to set in advance the closeness to the distributions of interest. The distribution of the stopping rule is derived, and the practicability of the simulating procedure is discussed. Sufficient conditions for convergence of random functionals are provided. The numerical applications provided just sketch the idea of the variety of nonparametric procedures that can be easily and safely implemented in a Bayesian setting.

Characterization of a Marshall-Olkin type class of distributions

SummaryA class of bivariate distributions that generalize Marshall-Olkins one is characterized through a functional equation which involves two associative operations. The obtained distributions concentrate positive mass on the linex=y when the two associative operations coincide; otherwise a positive mass is concentrated on a continuous monotone function.

Urn schemes and reinforced random walks

We define a reinforced urn process (RUP) to be a reinforced random walk on a state space of urns and we show its partial exchangeability. When it is recurrent, a RUP is a mixture of Markov chains and we characterize its mixing distribution on the space of stochastic matrices. Many Bayesian nonparametric priors, like Polya trees, the beta-Stacy process and, in general, neutral to the right processes can be derived from RUPs. Applications to survival data are examined.

A bayesian predictive approach to sequential search for an optimal dose: Parametric and nonparametric models

This paper looks at a new approach to the problem of finding the maximal tolerated dose (or optimal dose, Eichhorn and Zacks, 1973) of certain drugs which in addition to their therapeutic effects have secondary harmful effects.

A Bayesian non-parametric approach to survival analysis using Polya trees

This paper presents a Bayesian non-parametric approach to survival analysis based on arbitrarily right censored data. The analysis is based on posterior predictive probabilities using a Polya tree prior distribution on the space of probability measures on [0, ∞). In particular we show that the estimate generalizes the classical Kaplanndash;Meier non-parametric estimator, which is obtained in the limiting case as the weight of prior information tends to zero.

A bivariate Dirichlet process

This paper introduces a bivariate Dirichlet process for modelling a partially exchangeable sequence of observables. The proposed model would be relevant when two distributions are unknown but are thought to be close to each other. For two random distributions with the same marginals, the belief in the degree of closeness is expressed through the correlation between masses assigned to equal sets.

Mixtures of products of Dirichlet processes for variable selection in survival analysis

A very important problem in survival analysis is the accurate selection of the relevant prognostic explanatory variables. We propose a novel approach, based on mixtures of products of Dirichlet process priors, that provides a formal inferential tool to compare the explanatory power of each covariate, in terms of the marginal likelihood attached to the induced partitions of the observations. Our proposed model is Bayesian nonparametric, and, thus, keeps the amount of model specification to a minimum, increasing robustness of the final inferences.

Sequential Testing Problems for Lévy Processes

Abstract We present the sequential testing of two simple hypotheses for a large class of Lévy processes. As usual in this framework, the initial optimal stopping problem is reduced to a free-boundary problem, solved through the principles of the smooth and/or continuous fit. The well-known solutions of the Wiener and the Poisson sequential testing can be derived from our procedure. The exact solution for sequentially testing two simple hypotheses concerning the parameter p, 0 < p < 1, of a negative binomial process is explicitly given.

The Gini concentration test for survival data

We apply the well known Gini index to the measurement of concentration in survival times within groups of patients, and as a way to compare the distribution of survival times across groups of patients in clinical studies. In particular, we propose an estimator of a restricted version of the index from right censored data. We derive the asymptotic distribution of the resulting Gini statistic, and construct an estimator for its asymptotic variance. We use these results to propose a novel test for differences in the heterogeneity of survival distributions, which may suggest the presence of a differential treatment effect for some groups of patients. We focus in particular on traditional and generalized cure rate models, i.e., mixture models with a distribution of the lifetimes of the cured patients that is either degenerate at infinity or has a density. Results from a simulation study suggest that the Gini index is useful in some situations, and that it should be considered together with existing tests (in particular, the Log-rank, Wilcoxon, and Gray–Tsiatis tests). Use of the test is illustrated on the classic data arising from the Eastern Cooperative Oncology Group melanoma clinical trial E1690.