Distributional results for means of normalized random measures with independent increments. (English) Zbl 1068.62034

The authors consider random measures \(\tilde\phi\) generated by increasing processes with independent increments using a time change. The introduced concept generalises Dirichlet random probability measures that are obtained using a time change of a reparametrised gamma-subordinator whose increments follow the gamma distribution. An expression for the distribution of the mean of the introduced random measure is found by using the Gurland inversion formula for characteristic functions.
Given a sample \(X_1,\dots,X_n\) of random variables, conditionally i.i.d. given the random measure \(\tilde\phi\) and distributed according to \(\tilde\phi\), the authors derive an expression for the posterior distribution of the mean of \(\tilde\phi\) which is useful for the Bayesian inference. A lengthy final section presents two examples of statistical relevance.


62F15 Bayesian inference
60G57 Random measures
60G51 Processes with independent increments; Lévy processes
Full Text: DOI


[1] BARLOW, M., PITMAN, J. and YOR, M. (1989). Une extension multidimensionnelle de la loi de l’arc sinus. Séminaire de Probabilités XXIII. Lecture Notes in Math. 1372 294-314. Springer, Berlin. · Zbl 0738.60072
[2] BILLINGSLEY, P. (1995). Probability and Measure, 3rd ed. Wiley, New York. · Zbl 0822.60002
[3] BILLINGSLEY, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York. · Zbl 0944.60003
[4] BILODEAU, M. and BRENNER, D. (1999). Theory of Multivariate Statistics. Springer, New York. · Zbl 0930.62054
[5] CIFARELLI, D. M. and REGAZZINI, E. (1990). Distribution functions of means of a Dirichlet process. Ann. Statist. 18 429-442. [Correction (1994) 22 1633-1634.] · Zbl 0706.62012
[6] CIFARELLI, D. M. and REGAZZINI, E. (1996). Tail-behaviour and finiteness of means of distributions chosen from a Dirichlet process. Technical report IAMI-CNR.
[7] DAMIEN, P., LAUD, P. W. and SMITH, A. F. M. (1995). Approximate random variate generation from infinitely divisible distributions with applications to Bayesian inference. J. Roy. Statist. Soc. Ser. B 57 547-563. JSTOR: · Zbl 0827.60009
[8] DOKSUM, K. (1974). Tailfree and neutral random probabilities and their posterior distributions. Ann. Probab. 2 183-201. · Zbl 0279.60097
[9] FEIGIN, P. D. and TWEEDIE, R. L. (1989). Linear functionals and Markov chains associated with Dirichlet processes. Math. Proc. Cambridge Philos. Soc. 105 579-585. · Zbl 0677.60080
[10] FERGUSON, T. S. (1974). Prior distributions on spaces of probability measures. Ann. Statist. 2 615- 629. · Zbl 0286.62008
[11] FERGUSON, T. S. and PHADIA, E. G. (1979). Bayesian nonparametric estimation based on censored data. Ann. Statist. 7 163-186. · Zbl 0401.62031
[12] GRADSHTEy N, I. S. and Ry ZHIK, I. M. (2000). Table of Integrals, Series, and Products, 6th ed. Academic Press, New York.
[13] GURLAND, J. (1948). Inversion formulae for the distributions of ratios. Ann. Math. Statist. 19 228- 237. · Zbl 0032.03403
[14] HJORT, N. L. (1990). Nonparametric Bay es estimators based on beta processes in models for life history data. Ann. Statist. 18 1259-1294. · Zbl 0711.62033
[15] KINGMAN, J. F. C. (1967). Completely random measures. Pacific J. Math. 21 59-78. · Zbl 0155.23503
[16] KINGMAN, J. F. C. (1975). Random discrete distributions (with discussion). J. Roy. Statist. Soc. Ser. B 37 1-22. JSTOR: · Zbl 0331.62019
[17] KINGMAN, J. F. C. (1993). Poisson Processes. Oxford Univ. Press. · Zbl 0771.60001
[18] MORANDO, P. (1969). Mesures aléatoires. Séminaire Probabilité de Université Strasbourg 3 190- 229. · Zbl 0183.45802
[19] OLDHAM, K. B. and SPANIER, J. (1974). The Fractional Calculus. Academic Press, New York. · Zbl 0292.26011
[20] PERMAN, M. (1993). Order statistics for jumps of normalized subordinators. Stochastic Process. Appl. 46 267-281. · Zbl 0777.60070
[21] PITMAN, J. (1996). Some developments of the Blackwell-MacQueen urn scheme. In Statistics, Probability and Game Theory. Papers in Honor of David Blackwell (T. S. Ferguson, L. S. Shapley and J. B. MacQueen, eds.). Lecture Notes-Monograph Series 30 245-267.
[22] IMS, Hay ward, CA.
[23] PITMAN, J. and YOR, M. (1997). The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 855-900. · Zbl 0880.60076
[24] PRUDNIKOV, A. P., BRy CHKOV, YU. A. and MARICHEV, O. I. (1992). Integrals and Series: Direct Laplace Transforms 4. Gordon and Breach, New York. · Zbl 0786.44003
[25] REGAZZINI, E. (1978). Intorno ad alcune questioni relative alla definizione del premio secondo la teoria della credibilità. Giorn. Ist. Ital. Attuari 41 77-89. · Zbl 0415.62082
[26] REGAZZINI, E., GUGLIELMI, A. and DI NUNNO, G. (2002). Theory and numerical analysis for exact distributions of functionals of a Dirichlet process. Ann. Statist. 30 1376-1411. · Zbl 1018.62011
[27] REGAZZINI, E. and SAZONOV, V. V. (2001). Approximation of laws of random probabilities by mixtures of Dirichlet distributions with applications to nonparametric Bayesian inference. Theory Probab. Appl. 45 93-110. · Zbl 0984.60030
[28] SATO, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press. · Zbl 0973.60001
[29] SKOROHOD, A. V. (1991). Random Processes with Independent Increments. Kluwer, Dordrecht. · Zbl 0732.60081
[30] TSILEVICH, N., VERSHIK, A. and YOR, M. (2001). An infinite-dimensional analogue of the Lebesgue measure and distinguished properties of the gamma process. J. Funct. Anal. 185 274-296. · Zbl 0990.60053
[31] WALKER, S. and DAMIEN, P. (1998). A full Bayesian non-parametric analysis involving a neutral to the right process. Scand. J. Statist. 25 669-680.
[32] WALKER, S. and MULIERE, P. (1997). Beta-Stacy processes and a generalization of the Póly a-urn scheme. Ann. Statist. 25 1762-1780. · Zbl 0928.62067
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.