Electronic Communications in Probability

Distribution of a random functional of a Ferguson-Dirichlet process over the unit sphere

Thomas Jiang and Kun-Lin Kuo

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Abstract

Jiang, Dickey, and Kuo [12] gave the multivariate c-characteristic function and showed that it has properties similar to those of the multivariate Fourier transformation. We first give the multivariate c-characteristic function of a random functional of a Ferguson-Dirichlet process over the unit sphere. We then find out its probability density function using properties of the multivariate c-characteristic function. This new result would generalize that given by [11].

Article information

Source
Electron. Commun. Probab., Volume 13 (2008), paper no. 49, 518-525.

Dates
Accepted: 14 October 2008
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465233476

Digital Object Identifier
doi:10.1214/ECP.v13-1416

Mathematical Reviews number (MathSciNet)
MR2447838

Zentralblatt MATH identifier
1192.60040

Subjects
Primary: 60E10: Characteristic functions; other transforms
Secondary: 62E15: Exact distribution theory

Keywords
Ferguson-Dirichlet process c-characteristic function

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Jiang, Thomas; Kuo, Kun-Lin. Distribution of a random functional of a Ferguson-Dirichlet process over the unit sphere. Electron. Commun. Probab. 13 (2008), paper no. 49, 518--525. doi:10.1214/ECP.v13-1416. https://projecteuclid.org/euclid.ecp/1465233476


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References

  • B.C. Carlson. Special Functions of Applied Mathematics. Academic Press, New York, 1977.
  • D.M. Cifarelli, E. Regazzini. Distribution functions of means of a Dirichlet process. Ann. Statist. 18 (1990), 429–442. Correction: Ann. Statist. 22 (1994), 1633–1634.
  • P. Diaconis, J. Kemperman. Some new tools for Dirichlet priors, in: J.M. Bernardo, J.O. Berger, A.P. Dawid, and A.F.M. Smith eds. Bayesian Statistics 5, pp. 97–106. Oxford University Press, (1994).
  • I. Epifani, A. Guglielmi, E. Melilli. A stochastic equation for the law of the random Dirichlet variance. Statist. Probab. Lett. 76 (2006), 495–502.
  • T.S. Ferguson. A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 (1973), 209–230.
  • I.S. Gradshteyn, I.M. Ryzhik. Table of Integrals, Series, and Products. Academic Press, New York, 6th ed, 2000.
  • W. Gröbner, W. Hofreiter. Integraltafel, Vol. 2. Springer-Verlag, New York, 5th ed, 1973.
  • R.C. Hannum, M. Hollander, N.A. Langberg. Distributional results for random functionals of a Dirichlet process. Ann. Probab. 9 (1981), 665–670.
  • N.L. Hjort, A. Ongaro. Exact inference for random Dirichlet means. Stat. Inference Stoch. Process. 8 (2005), 227–254.
  • J. Jiang. Starlike functions and linear functions of a Dirichlet distributed vector. SIAM J. Math. Anal. 19 (1988), 390–397.
  • T.J. Jiang. Distribution of random functional of a Dirichlet process on the unit disk. Statist. Probab. Lett. 12 (1991), 263–265.
  • T.J. Jiang, J.M. Dickey, K.-L. Kuo. A new multivariate transform and the distribution of a random functional of a Ferguson-Dirichlet process. Stochastic Process. Appl. 111 (2004), 77–95.
  • T.J. Jiang, K.-L. Kuo. On the random functional of the Ferguson-Dirichlet process. 2006 Proceeding of the Section on Bayesian Statistical Science of the American Statistical Association, pp. 52–59.
  • A. Lijoi, A E. Regazzini. Means of a Dirichlet process and multiple hypergeometric functions. Ann. Probab. 32 (2004), 1469–1495.
  • A.Y. Lo. On a class of Bayesian nonparametric estimates: I. density estimates. Ann. Statist. 12 (1984), 351–357.
  • E. Regazzini, A. Guglielmi, G. Di Nunno. Theory and numerical analysis for exact distributions of functionals of a Dirichlet process. Ann. Statist. 30 (2002), 1376–1411.
  • H. Yamato, H. Characteristic functions of means of distributions chosen from a Dirichlet process. Ann. Probab. 12 (1984), 262–267.