The Annals of Statistics

Normalized random measures driven by increasing additive processes

Luis E. Nieto-Barajas, Igor Prünster, and Stephen G. Walker

Full-text: Open access

Abstract

This paper introduces and studies a new class of nonparametric prior distributions. Random probability distribution functions are constructed via normalization of random measures driven by increasing additive processes. In particular, we present results for the distribution of means under both prior and posterior conditions and, via the use of strategic latent variables, undertake a full Bayesian analysis. Our class of priors includes the well-known and widely used mixture of a Dirichlet process.

Article information

Source
Ann. Statist., Volume 32, Number 6 (2004), 2343-2360.

Dates
First available in Project Euclid: 7 February 2005

Permanent link to this document
https://projecteuclid.org/euclid.aos/1107794871

Digital Object Identifier
doi:10.1214/009053604000000625

Mathematical Reviews number (MathSciNet)
MR2153987

Zentralblatt MATH identifier
1069.62029

Subjects
Primary: 62F15: Bayesian inference
Secondary: 60G57: Random measures

Keywords
Bayesian nonparametric inference distribution of means of random probability measures increasing additive process Lévy measure mixtures of Dirichlet process

Citation

Nieto-Barajas, Luis E.; Prünster, Igor; Walker, Stephen G. Normalized random measures driven by increasing additive processes. Ann. Statist. 32 (2004), no. 6, 2343--2360. doi:10.1214/009053604000000625. https://projecteuclid.org/euclid.aos/1107794871


Export citation

References

  • Barndorff-Nielsen, O. E. and Shephard, N. (2001). Non-Gaussian Ornstein--Uhlenbeck-based models and some of their uses in financial economics. J. R. Stat. Soc. Ser. B Stat. Methodol. 63 167--241.
  • Brockwell, P. J. (2001). Lévy driven CARMA processes. Nonlinear non-Gaussian models and related filtering methods. Ann. Inst. Statist. Math. 53 113--124.
  • 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. Parte II. Riv. Mat. Sci. Econom. Social. 2 95--111.
  • 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.]
  • Cifarelli, D. M. and Regazzini, E. (1996). Tail-behaviour and finiteness of means of distributions chosen from a Dirichlet process. Technical Report IAMI-CNR 96.19.
  • Dey, D., Müller, P. and Sinha, D., eds. (1998). Practical Nonparametric and Semiparametric Bayesian Statistics. Lecture Notes in Statist. 133. Springer, New York.
  • Dykstra, R. L. and Laud, P. W. (1981). A Bayesian nonparametric approach to reliability. Ann. Statist. 9 356--367.
  • Epifani, I., Lijoi, A. and Prünster, I. (2003). Exponential functionals and means of neutral-to-the-right priors. Biometrika 90 791--808.
  • Escobar, M. D. (1988). Estimating the means of several normal populations by nonparametric estimation of the distribution of the means. Ph.D. dissertation, Dept. Statistics, Yale Univ.
  • Escobar, M. D. and West, M. (1995). Bayesian density estimation and inference using mixtures. J. Amer. Statist. Assoc. 90 577--588.
  • 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.
  • Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 209--230.
  • Ferguson, T. S. and Klass, M. J. (1972). A representation of independent increment processes without Gaussian components. Ann. Math. Statist. 43 1634--1643.
  • Gill, R. D. and Johansen, S. (1990). A survey of product integration with a view toward application in survival analysis. Ann. Statist. 18 1501--1555.
  • 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.
  • Gurland, J. (1948). Inversion formulae for the distributions of ratios. Ann. Math. Statist. 19 228--237.
  • Hjort, N. L. (1990). Nonparametric Bayes estimators based on beta processes in models for life history data. Ann. Statist. 18 1259--1294.
  • Hjort, N. L. (2003). Topics in nonparametric Bayesian statistics. In Highly Structured Stochastic Systems (P. J. Green, N. L. Hjort and S. Richardson, eds.) 455--487. Oxford Univ. Press.
  • Lijoi, A. and Regazzini, E. (2004). Means of a Dirichlet process and multiple hypergeometric functions. Ann. Probab. 32 1469--1495.
  • Lo, A. Y. (1984). On a class of Bayesian nonparametric estimates. I. Density estimates. Ann. Statist. 12 351--357.
  • MacEachern, S. N. and Müller, P. (1998). Estimating mixture of Dirichlet process models. J. Comput. Graph. Statist. 7 223--238.
  • Muliere, P. and Tardella, L. (1998). Approximating distributions of random functionals of Ferguson--Dirichlet priors. Canad. J. Statist. 26 283--297.
  • Nieto-Barajas, L. E. and Walker, S. G. (2004). Bayesian nonparametric survival analysis via Lévy driven Markov processes. Statist. Sinica 14 1127--1146.
  • 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.
  • Regazzini, E., Lijoi, A. and Prünster, I. (2003). Distributional results for means of normalized random measures with independent increments. Ann. Statist. 31 560--585.
  • Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press.
  • Walker, S. G. and Damien, P. (2000). Representations of Lévy processes without Gaussian components. Biometrika 87 477--483.
  • Walker, S. G. and Muliere, P. (1997). Beta-Stacy processes and a generalization of the Pólya-urn scheme. Ann. Statist. 25 1762--1780.
  • Wolpert, R. L. and Ickstadt, K. (1998). Poisson/gamma random field models for spatial statistics. Biometrika 85 251--267.
  • Wolpert, R. L., Ickstadt, K. and Hansen, M. B. (2003). A nonparametric Bayesian approach to inverse problems. In Bayesian Statistics 7 (J. M. Bernardo, M. J. Bayarri, J. O. Berger, A. P. Dawid, D. Heckerman, A. F. M. Smith and M. West, eds.) 403--418. Oxford Univ. Press.