A Berry-Esseen theorem for Pitman's α\alpha-diversity

This paper is concerned with the study of the random variable $K_n$ denoting the number of distinct elements in a random sample $(X_1, \dots, X_n)$ of exchangeable random variables driven by the two parameter Poisson-Dirichlet distribution, $PD(\alpha,\theta)$. For $\alpha\in(0,1)$, Theorem 3.8 in \cite{Pit(06)} shows that $\frac{K_n}{n^{\alpha}}\stackrel{\text{a.s.}}{\longrightarrow} S_{\alpha,\theta}$ as $n\rightarrow+\infty$. Here, $S_{\alpha,\theta}$ is a random variable distributed according to the so-called scaled Mittag-Leffler distribution. Our main result states that \sup_{x \geq 0} \Big| \ppsf\Big[\frac{K_n}{n^{\alpha}} \leq x \Big] - \ppsf[S_{\alpha,\theta} \leq x] \Big| \leq \frac{C(\alpha, \theta)}{n^{\alpha}} holds with an explicit constant $C(\alpha, \theta)$. The key ingredients of the proof are a novel probabilistic representation of $K_n$ as compound distribution and new, refined versions of certain quantitative bounds for the Poisson approximation and the compound Poisson distribution

Large deviation principles for the Ewens-Pitman sampling model

Let $M_{l,n}$ be the number of blocks with frequency $l$ in the exchangeable random partition induced by a sample of size $n$ from the Ewens-Pitman sampling model. We show that, as $n$ tends to infinity, $n^{-1}M_{l,n}$ satisfies a large deviation principle and we characterize the corresponding rate function. A conditional counterpart of this large deviation principle is also presented. Specifically, given an initial sample of size $n$ from the Ewens-Pitman sampling model, we consider an additional sample of size $m$. For any fixed $n$ and as $m$ tends to infinity, we establish a large deviation principle for the conditional number of blocks with frequency $l$ in the enlarged sample, given the initial sample. Interestingly, the conditional and unconditional large deviation principles coincide, namely there is no long lasting impact of the given initial sample. Potential applications of our results are discussed in the context of Bayesian nonparametric inference for discovery probabilities.Comment: 30 pages, 2 figure

Approximating predictive probabilities of Gibbs-type priors

Gibbs-type random probability measures, or Gibbs-type priors, are arguably the most "natural" generalization of the celebrated Dirichlet prior. Among them the two parameter Poisson-Dirichlet prior certainly stands out for the mathematical tractability and interpretability of its predictive probabilities, which made it the natural candidate in several applications. Given a sample of size $n$, in this paper we show that the predictive probabilities of any Gibbs-type prior admit a large $n$ approximation, with an error term vanishing as $o(1/n)$, which maintains the same desirable features as the predictive probabilities of the two parameter Poisson-Dirichlet prior.Comment: 22 pages, 6 figures. Added posterior simulation study, corrected typo

Group versus individual discrimination among young workers: a distributional approach

We evaluate the gender wage gap and the unexplained gender wage differential for workers 15-29 year old during the period 1990-1997, using a particularly rich set of data from the Italian Social Security System covering all individuals in the labour markets of two Italian provinces. We estimate separate earnings functions for men and women correcting for endogeneity of education and we evaluate gender discrimination by studying the entire distribution of the unexplained wage gap as suggested by Jenkins (1994). We evaluate discrimination against females by means of bivariate density functions. This innovation makes it possible to condition the density distribution on the marginal distribution of any characteristic and to evaluate more precisely the existence of group and individual discrimination. Our analysis suggests that discrimination is not evenly distributed among women, in relation to their characteristics; in particular, there is evidence of lower discrimination against highly educated females. Moreover in 1997, compared to 1990, discrimination increased in a appreciable way, affecting human capital rich females more significantly. While our work is based in a very local context the richness of the data and the methodological innovation give the results a wider application.wage differentials, wage discrimination, gender

Bayesian nonparametric analysis of reversible Markov chains

We introduce a three-parameter random walk with reinforcement, called the $(\theta,\alpha,\beta)$ scheme, which generalizes the linearly edge reinforced random walk to uncountable spaces. The parameter $\beta$ smoothly tunes the $(\theta,\alpha,\beta)$ scheme between this edge reinforced random walk and the classical exchangeable two-parameter Hoppe urn scheme, while the parameters $\alpha$ and $\theta$ modulate how many states are typically visited. Resorting to de Finetti's theorem for Markov chains, we use the $(\theta,\alpha,\beta)$ scheme to define a nonparametric prior for Bayesian analysis of reversible Markov chains. The prior is applied in Bayesian nonparametric inference for species sampling problems with data generated from a reversible Markov chain with an unknown transition kernel. As a real example, we analyze data from molecular dynamics simulations of protein folding.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1102 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Asymptotics for a Bayesian nonparametric estimator of species variety

In Bayesian nonparametric inference, random discrete probability measures are commonly used as priors within hierarchical mixture models for density estimation and for inference on the clustering of the data. Recently, it has been shown that they can also be exploited in species sampling problems: indeed they are natural tools for modeling the random proportions of species within a population thus allowing for inference on various quantities of statistical interest. For applications that involve large samples, the exact evaluation of the corresponding estimators becomes impracticable and, therefore, asymptotic approximations are sought. In the present paper, we study the limiting behaviour of the number of new species to be observed from further sampling, conditional on observed data, assuming the observations are exchangeable and directed by a normalized generalized gamma process prior. Such an asymptotic study highlights a connection between the normalized generalized gamma process and the two-parameter Poisson-Dirichlet process that was previously known only in the unconditional case.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ371 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Brevi spunti di riflessione sull’attuale valenza del rapporto comunicativo tra diritto pubblico e diritto privato

Il rapporto tra la sfera pubblica e la sfera privata dell’esperienza giuridica italiana contemporanea, alla luce della normativa costituzionale e ordinaria, non è riduzione del diritto privato nel diritto pubblico, né, viceversa, riduzione del diritto pubblico a esplicitazione di interessi privati, ma deve concepirsi come insopprimibile e dialogica compresenza dinamica, e come comunicazione dialettica, dei due termini, pena la dissoluzione dell’unità dell’ordinamento

La teoria della complessità sociale e il postmoderno nel diritto. Un esempio paradigmatico: la teoria della rappresentanza di Salvatore Pugliatti

Il pensiero giuridico e la teoria generale della rappresentanza elaborati da Salvatore Pugliatti costituiscono, ante litteram, un rilevante e paradigmatico contributo per la chiarificazione, anche in ambito giuridico, del concetto contemporaneo di postmodernità, anticipando, e costituendo già estrinsecazione, dell’ineliminabile rapporto comunicativo e dialogante sussistente, per la teoria della complessità sociale e del molteplice, tra la concretezza e la pluralità delle istanze storiche e l’astrattezza e l’unitarietà dei concetti che le cristallizzano in sistemi di pensiero

L'esperienza giuridica digitale tra teoria e prassi: spunti per una riflessione critica

Stefano Favaro "L'esperienza digitale tra teoria e prassi: spunti per una riflessione critica" in: Tigor: rivista di scienze della comunicazione. A.I (2009) n.1 (gennaio-giugno), pp. 71-87La rivoluzione giuridico-informatica del terzo millennio, sia sotto il profilo del diritto positivo dell’informatica che sotto il versante documentario e cognitivo, tratteggia l’esperienza giuridica digitale contemporanea come quella dimensione “computercentrica” le cui aporie, peraltro, sono superabili solo attraverso il recupero – paradossale ma imprescindibile – proprio della centralità, filosoficamente classica, dell’uomo