Theory and numerical analysis for exact distributions of functionals of a Dirichlet process. (English) Zbl 1018.62011

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

Ann. Stat. 30, No. 5, 1376-1411 (2002).

Summary: The distribution of a mean or, more generally, of a vector of means of a Dirichlet process is considered. Some characterizing aspects of this paper are: (i) a review of a few basic results, providing new formulations free from many of the extra assumptions considered to date in the literature, and giving essentially new, simpler and more direct proofs; (ii) new numerical evaluations, with any prescribed error of approximation, of the distribution at issue; (iii) a new form for the law of a vector of means.
Moreover, as applications of these results, we give: (iv) the sharpest condition sufficient for the distribution of a mean to be symmetric; (v) forms for the probability distribution of the variance of the Dirichlet random measure; (vi) some hints for determining the finite-dimensional distributions of a random function connected with Bayesian methods for queuing models.

MSC:

62E15	Exact distribution theory in statistics
65C60	Computational problems in statistics (MSC2010)
62F15	Bayesian inference
62E17	Approximations to statistical distributions (nonasymptotic)

Keywords:

Dirichlet process; distribution of vectors of linear functionals; numerical approximation of exact distributions

References:

[1]	ANDREWS, G. E., ASKEY, R. and ROY, R. (1999). Special Functions. Cambridge Univ. Press. · Zbl 0920.33001
[2]	BILODEAU, M. and BRENNER, D. (1999). Theory of Multivariate Statistics. Springer, New York. · Zbl 0930.62054
[3]	CIFARELLI, D. M. and MELILLI, E. (2000). Some new results for Dirichlet priors. Ann. Statist. 28 1390-1413. · Zbl 1105.62303 · doi:10.1214/aos/1015957399
[4]	CIFARELLI, D. M. and REGAZZINI, E. (1978). Problemi statistici non parametrici in condizioni di scambiabilità parziale: impiego di medie associative. Quaderni dell’Istituto di Matematica Finanziaria dell’Università di Torino Ser. III 1-36.
[5]	CIFARELLI, D. M. and REGAZZINI, E. (1979a). Considerazioni generali sull’impostazione Bayesiana di problemi non parametrici. Le medie associative nel contesto del processo aleatorio di Dirichlet I. Riv. Mat. Sci. Econom. Social. 2 39-52. · Zbl 0492.62041 · doi:10.1007/BF02626108
[6]	CIFARELLI, D. M. and REGAZZINI, E. (1979b). Considerazioni generali sull’impostazione Bayesiana di problemi non parametrici. Le medie associative nel contesto del processo aleatorio di Dirichlet II. Riv. Mat. Sci. Econom. Social. 2 95-111. · Zbl 0492.62042 · doi:10.1007/BF02620300
[7]	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 · doi:10.1214/aos/1176347509
[8]	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.
[9]	CIFARELLI, D. M. and REGAZZINI, E. (1996). Tail-behaviour and finiteness of means of distributions chosen from a Dirichlet process. Quaderno IAMI 96.19, CNR-IAMI, Milano.
[10]	CONTI, P. L. (1999). Large sample Bayesian analysis for Geo/G/1 discrete-time queuing models. Ann. Statist. 27 1785-1807. · Zbl 0963.62092 · doi:10.1214/aos/1017939239
[11]	DAVIES, R. B. (1973). Numerical inversion of a characteristic function. Biometrika 60 415-417. JSTOR: · Zbl 0263.65115 · doi:10.1093/biomet/60.2.415
[12]	DAWSON, D. A. (1993). Measure-valued Markov processes. École d’Été de Probabilités de St.-Flour XXI. Lecture Notes in Math. 1541 1-260. Springer, New York. · Zbl 0799.60080
[13]	DIACONIS, P. and KEMPERMAN, J. (1996). Some new tools for Dirichlet priors. In Bayesian Statistics 5 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.) 97-106. Clarendon Press, Oxford.
[14]	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 · doi:10.1017/S0305004100077951
[15]	FLORENS, J. P. and ROLIN, J.-M. (1994). Bay es, bootstrap, moments. Discussion paper 94.13, Institut de Statistique, Univ. Catholique de Louvain.
[16]	FOLLAND, G. B. (1984). Real Analy sis. Wiley, New York.
[17]	GELFAND, A. E. and KOTTAS, A. (2002). A computational approach for full nonparametric Bayesian inference under Dirichlet process mixture models J. Comput. Graph. Statist. 11 284-305. JSTOR: · doi:10.1198/106186002760180518
[18]	GELFAND, A. E. and MUKHOPADHy AY, S. (1995). On nonparametric Bayesian inference for the distribution of a random sample. Canad. J. Statist. 23 411-420. JSTOR: · Zbl 0858.62028 · doi:10.2307/3315384
[19]	GRADSHTEy N, I. S. and Ry ZHIK, I. M. (1994). Table of Integrals, Series, and Products, 5th ed. Academic Press, San Diego.
[20]	GUGLIELMI, A. (1998). A simple procedure calculating the generalized Stieltjes transform of the mean of a Dirichlet process. Statist. Probab. Lett. 38 299-303. · Zbl 0916.60046 · doi:10.1016/S0167-7152(98)00034-0
[21]	GUGLIELMI, A., HOLMES, C. and WALKER, S. (2002). Perfect simulation involving functions of a Dirichlet process. J. Comput. Graph. Statist. 11 306-310. JSTOR: · doi:10.1198/106186002760180527
[22]	GUGLIELMI, A. and TWEEDIE, R. (2001). Markov chain Monte Carlo estimation of the law of the mean of a Dirichlet process. Bernoulli 7 573-592. · Zbl 1005.62073 · doi:10.2307/3318726
[23]	GURLAND, J. (1948). Inversion formulae for the distributions of ratios. Ann. Math. Statist. 19 228- 237. · Zbl 0032.03403 · doi:10.1214/aoms/1177730247
[24]	HANNUM, R. C., HOLLANDER, M. and LANGBERG, N. A. (1981). Distributional results for random functionals of a Dirichlet process. Ann. Probab. 9 665-670. · Zbl 0473.60019 · doi:10.1214/aop/1176994373
[25]	HUGHETT, P. (1998). Error bounds for numerical inversion of a probability characteristic function. SIAM J. Numer. Anal. 35 1368-1392. JSTOR: · Zbl 0912.65119 · doi:10.1137/S003614299631085X
[26]	KALLENBERG, O. (1997). Foundations of Modern Probability. Springer, New York. · Zbl 0892.60001
[27]	LIJOI, A. and REGAZZINI, E. (2001). Means of a Dirichlet process and multiple hy pergeometric functions. Quaderno IAMI 01.09, CNR-IAMI, Milano.
[28]	MULIERE, P. and TARDELLA, L. (1998). Approximating distributions of random functionals of Ferguson-Dirichlet priors. Canad. J. Statist. 26 283-297. JSTOR: · Zbl 0913.62010 · doi:10.2307/3315511
[29]	OLDHAM, K. B. and SPANIER, J. (1974). The Fractional Calculus. Academic Press, New York. · Zbl 0292.26011
[30]	REGAZZINI, E. (1998). An example of the interplay between statistics and special functions. In Atti dei Convegni Lincei 147: Tricomi’s Ideas and Contemporary Applied Mathematics 303- 320. Accademia Nazionale dei Lincei, Roma. · Zbl 1136.33301
[31]	REGAZZINI, E., GUGLIELMI, A. and DI NUNNO, G. (2000). Theory and numerical analysis for exact distributions of functionals of a Dirichlet process. Quaderno IAMI 00.12, CNR-IAMI, Milano. · Zbl 1018.62011
[32]	REGAZZINI, E., LIJOI, A. and PRÜNSTER, I. (2003). Distributional results for means of normalized random measures with independent increments. Ann. Statist. 31. · Zbl 1068.62034 · doi:10.1214/aos/1051027881
[33]	TAMURA, H. (1988). Estimation of rare errors using expert judgment. Biometrika 75 1-9. JSTOR: · Zbl 0644.62033 · doi:10.1093/biomet/75.1.1
[34]	TSILEVICH, N., VERSHIK, A. and YOR, M. (2000). Distinguished properties of the gamma process and related topics. Prépublication 575, Laboratoire de Probabilité et Modeles aléatoires, Univ. Paris VI et Paris VII.
[35]	WALKER, S. G., DAMIEN, P., LAUD, P. W. and SMITH, A. F. M. (1999). Bayesian nonparametric inference for random distributions and related functions (with discussion). J. Roy. Statist. Soc. Ser. B 61 485-527. JSTOR: · Zbl 0983.62027 · doi:10.1111/1467-9868.00190
[36]	VIA FERRATA, 1 27100 PAVIA ITALY E-MAIL: [e-mail] [e-mail] A. GUGLIELMI ISTITUTO PER LE APPLICAZIONI DELLA MATEMATICA E DELL’INFORMATICA CONSIGLIO NAZIONALE DELLE RICERCHE VIA AMPERE, 56 20131 MILANO ITALY E-MAIL: [e-mail]