Theory and numerical analysis for exact distributions of functionals of a Dirichlet process

Regazzini, Eugenio; Guglielmi, Alessandra; Nunno, Giulia Di

doi:10.1214/aos/1035844980

Cited by 55 publications

(71 citation statements)

References 22 publications

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“…The same representation has been achieved in Proposition 9(iii) of [72] by removing this condition on α. The proof is based on a different approach and makes use of a simple and useful idea given in [33].…”

Section: Probability Distribution Of Random Dirichlet Meansmentioning

confidence: 54%

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Distributional properties of means of random probability measures

Lijoi¹,

Prünster²

2009

Statist. Surv.

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Abstract:The present paper provides a review of the results concerning distributional properties of means of random probability measures. Our interest in this topic has originated from inferential problems in Bayesian Nonparametrics. Nonetheless, it is worth noting that these random quantities play an important role in seemingly unrelated areas of research. In fact, there is a wealth of contributions both in the statistics and in the probability literature that we try to summarize in a unified framework. Particular attention is devoted to means of the Dirichlet process given the relevance of the Dirichlet process in Bayesian Nonparametrics. We then present a number of recent contributions concerning means of more general random probability measures and highlight connections with the moment problem, combinatorics, special functions, excursions of stochastic processes and statistical physics.

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Section: Probability Distribution Of Random Dirichlet Meansmentioning

confidence: 54%

“…Moreover, α m converges weakly to α as m → ∞. This approach has been developed in [71] and later applied in [72]. A similar discretization procedure is also at the basis of the proof given in [36].…”

Section: Cifarelli-regazzini Identity and Related Resultsmentioning

confidence: 99%

Distributional properties of means of random probability measures

Lijoi¹,

Prünster²

2009

Statist. Surv.

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“…We assume that the values of the grid A are equally-spaced by .005, with minimum .005 and maximum ξ. We now consider a deterministic approach to evaluating the distribution of a functional (e.g., β * (F * )) of the MDP posterior, as in previous research on DP functionals (Regazzini, et al 2002). Specifically, here we employ the multivariate delta method to approximate the (MDP bootstrap) posterior distribution of β(F * ) via a Taylor series approximation β * (n * ) ≈ β(n * ) of β(n * ) around the mean, n * .…”

Section: Bootstrap Approximation To the Mdp Posterior Distributionmentioning

confidence: 99%

“…This has led researchers to propose various leverage-adjusted sandwich estimators (see MacKinnon, 2013). error of approximation (for reviews, see Regazzini et al, 2002;Lijoi & Prunster, 2009). This research has primarily focused the mean and other linear functionals of the DP.…”

Section: Introductionmentioning

confidence: 99%

A Dirichlet process functional approach to heteroscedastic-consistent covariance estimation

Karabatsos

2016

International Journal of Approximate Reasoning

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The mixture of Dirichlet process (MDP) defines a flexible prior distribution on the space of probability measures. This study shows that ordinary least-squares (OLS) estimator, as a functional of the MDP posterior distribution, has posterior mean given by weighted least-squares (WLS), and has posterior covariance matrix given by the (weighted) heteroscedastic-consistent sandwich estimator. This is according to a pairs bootstrap distribution approximation of the posterior, using a Pólya urn scheme. Also, when the MDP prior baseline distribution is specified as a product of independent probability measures, this WLS solution provides a new type of generalized ridge regression estimator. Such an estimator can handle multicollinear or singular design matrices even when the number of covariates exceeds the sample size, and can shrinks the coefficient estimates of irrelevant covariates towards zero, which makes it useful for nonlinear regressions via basis expansions. Also, this MDP/OLS functional methodology can be extended to methods for analyzing the sensitivity of the heteroscedasticity-consistent causal effect size over a range of hidden biases due to missing covariates omitted from the regression, and more generally extended to a Vibration of Effects analysis. The methodology is illustrated through the analysis of simulated and real data sets. Overall, this study establishes new connections between Dirichlet process functional inference, the bootstrap, consistent sandwich covariance estimation, ridge shrinkage regression, WLS, and sensitivity analysis, to provide regression methodology useful for inferences of the mean dependent response.

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“…Although, ψ dP has finite mean if and only if |ψ| dG < ∞, P has a significantly shorter tail than that of G. For instance, the random P generated by a Dirichlet process with Cauchy base measure has all moments. Distributions of the random mean functional has been studied in many articles including Cifarelli and Regazzini (1990) and Regazzini et al (2002). Interestingly the distribution of x dP (x) is G if and only if G is Cauchy.…”

Section: As P (A) Is Distributed As Beta (Mg(a) Mg(a C ))mentioning

confidence: 99%