The Annals of Statistics

Some new results for Dirichlet priors

Donato Michele Cifarelli and Eugenio Melilli

Full-text: Open access


Let p be a random probability measure chosen by a Dirichlet process whose parameter a is a finite measure with support contained in $[0, +\infty)$ and suppose that $V = \int x^2p(dx)-[\int xp(dx)]^2$ is a (finite)random variable. This paper deals with the distribution of $V$, which is given in a rather general case. A simple application to Bayesian bootstrap is also illustrated.

Article information

Ann. Statist., Volume 28, Number 5 (2000), 1390-1413.

First available in Project Euclid: 12 March 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G99: None of the above, but in this section 62E15: Exact distribution theory

Dirichlet process distribution of the variance hypergeometric functions


Cifarelli, Donato Michele; Melilli, Eugenio. Some new results for Dirichlet priors. Ann. Statist. 28 (2000), no. 5, 1390--1413. doi:10.1214/aos/1015957399.

Export citation


  • Basu, D. and Tiwari, R. C. (1982). A note on the Dirichlet process. In Statistics and Probability: Essays in Honor of C. R. Rao (G. Kallinpur, P. R. Krishnaiah and J. K. Ghosh, eds.) 89-103.
  • Berk, R. H. and Savage, I. R. (1979). Dirichlet processes produce discrete measures: an elementary proof. In Contributions to Statistics. Jaroslav Hajek Memorial Volume (J. Jureckova, ed.) 25-31. North-Holland, Prague.
  • Choudhuri, N. (1998). Bayesian bootstrap credible sets for multidimensional mean functional. Ann. Statist. 26 2104-2127.
  • Cifarelli, D. M. and Regazzini, E. (1979). Considerazioni generali sull'impostazione bayesiana di problemi non parametrici. Le medie associative nel contesto del processo aleatorio di Dirichlet I, II. Riv. Mat. Sci. Econom. Social 2 39-52.
  • Cifarelli, D. M. and Regazzini, E. (1990). Distribution functions of means of a Dirichlet process.
  • Ann. Statist. 18 429-442 (correction: Ann. Statist. 22 1633-1634).
  • Cifarelli, D. M. and Regazzini, E. (1993). Some remarks on the distribution functions of means of a Dirichlet process. Quaderno IAMI 93.4, CNR-IAMI, Milano.
  • Cifarelli, D. M. and Regazzini, E. (1995). Further remarks on the tails of a Ferguson-Dirichlet random probability distribution function. Quaderno IAMI 95.19, CNR-IAMI, Milano.
  • Diaconis, P. and Freedman, D. A. (1986). On the consistency of Bayes estimates. Ann. Statist. 14 1-26.
  • Diaconis, P. and Kemperma n, J. (1996). Some new tools for Dirichlet priors In Fifth International Meeting on Bayesian Statistics, Alicante.
  • Erdelyi, A. (1954). Tables of Integral Transforms. McGraw-Hill, New York.
  • Fabius, J. (1964). Asymptotic behaviour of Bayes estimate. Ann. Math. Statist. 35 846-856.
  • Feigin, P. D. and Tweedie, R. L. (1989). Linear functionals and Markov chains associated with Dirichlet processes. Math. Proc. Philos. Soc. 105 579-585.
  • Ferguson, T. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 209-230.
  • Freedman, D. A. (1963). On the asymptotic behaviour of Bayes' estimate in the discrete case. Ann. Math. Statist. 34 1386-1403.
  • Gasparini, F. (1995). Exact multivariate Bayesian bootstrap distributions of moments. Ann. Statist. 23 762-768.
  • Gradshteyn, I. S. and Ryzhik, I. M. (1980). Table of Integrals, Series, and Products. Academic Press, Boston.
  • Guglielmi, A. (1998). Numerical analysis for the distribution function of the mean of a Dirichlet process. Quaderno IAMI 98.01, CNR-IAMI, Milano.
  • Hannum, R. C., Hollander, M. and Langberg, N. A. (1981). Distributional results for random functionals of a Dirichlet process, Ann. Probab. 9 665-670.
  • Lo, A. Y. (1987). A larger sample study of the Bayesian bootstrap. Ann. Statist. 15 360-375.
  • Lu, I. and Richards, D. (1993). Random discriminants. Ann. Statist. 21 1982-2000.
  • Rubin, D. B. (1981). The Bayesian bootstrap. Ann. Statist. 9 130-134.
  • Sethuraman, J. and Tiwari, R. (1982). Convergence of Dirichlet measures and the interpretation of their parameter. In Statistical Decision Theory and Related Topics III, 2 305-315.
  • Yamato, H. (1984). Characteristic functions of means of distributions chosen from a Dirichlet process. Ann. Probab. 12 262-267.