Bayesian Analysis

On Asymptotic Properties and Almost Sure Approximation of the Normalized Inverse-Gaussian Process

Luai Al Labadi and Mahmoud Zarepour

Full-text: Open access

Abstract

In this paper, similar to the frequentist asymptotic theory, we present large sample theory for the normalized inverse-Gaussian process and its corresponding quantile process. In particular, when the concentration parameter is large, we establish the functional central limit theorem, the strong law of large numbers and the Glivenko-Cantelli theorem for the normalized inverse-Gaussian process and its related quantile process. We also derive a finite sum representation that converges almost surely to the Ferguson and Klass representation of the normalized inverse-Gaussian process. This almost sure approximation can be used to simulate the normalized inverse-Gaussian process.

Article information

Source
Bayesian Anal., Volume 8, Number 3 (2013), 553-568.

Dates
First available in Project Euclid: 9 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.ba/1378729919

Digital Object Identifier
doi:10.1214/13-BA821

Mathematical Reviews number (MathSciNet)
MR3102225

Zentralblatt MATH identifier
1329.60131

Keywords
Brownian bridge Dirichlet process Ferguson and Klass representation Nonparametric Bayesian inference Normalized inverse-Gaussian process Quantile process Weak convergence

Citation

Al Labadi, Luai; Zarepour, Mahmoud. On Asymptotic Properties and Almost Sure Approximation of the Normalized Inverse-Gaussian Process. Bayesian Anal. 8 (2013), no. 3, 553--568. doi:10.1214/13-BA821. https://projecteuclid.org/euclid.ba/1378729919


Export citation

References

  • Abramowitz, M. and Stegun, I. (1972). Handbook of mathematical functions with formulas, graphs, and mathematical tables. Mineola, New York: Dover Publications.
  • Al Labadi, L. and Zarepour, M. (2012). “On simulation from the two-parameter Poisson-Dirichlet process and the normalized inverse-Gaussian process.” Submitted.
  • Berti, P. and Rigo, P. (1997). “A Glivenko-Cantelli theorem for exchangeable random variables.” Statistics and Probability Letters, 32: 385–391.
  • Bickel, P. J. and Freedman, D. A. (1981). “Some asymptotic theory for the bootstrap.” The Annals of Statistics, 9: 1196–1217.
  • Bickel, P. J. and Wichura, M. J. (1971). “Convergence criteria for multiparameter stochastic processes and some applications.” The Annals of Mathematical Statistics, 42: 1656–1670.
  • Billingsley, P. (1995). Probability and Measure. New York: John Wiley & Sons, second edition.
  • — (1999). Convergence of Probability Measures. New York: John Wiley & Sons, third edition.
  • DasGupta, A. (2008). Asymptotic Theory of Statistics and Probability. New York: Springer.
  • Favaro, S., Lijoi, A., and Prünster, I. (2012). “On the stick-breaking representation of normalized inverse Gaussian priors.” Biometrika, 99: 663–774.
  • Favaro, S., Prünster, I., and Walker, S. G. (2011). “On a class of random probability measures with general predictive structure.” Scandinavian Journal of Statistics, 38: 359–376.
  • Ferguson, T. S. (1973). “A Bayesian Analysis of Some Nonparametric Problems.” The Annals of Statistics, 1: 209–230.
  • Ferguson, T. S. and Klass, M. J. (1972). “A Representation of independent increment processes without Gaussian components.” The Annals of Mathematical Statistics, 1: 209–230.
  • Ishwaran, H., James, L. F., and Zarepour, M. (2009). “An alternative to m out of n bootstrap.” Journal of Statistical Planning and Inference, 39: 788–801.
  • Ishwaran, H. and Zarepour, M. (2002). “Exact and approximate sum representations for the Dirichlet process.” Canadian Journal of Statistics, 30: 269–283.
  • James, L. F. (2008). “Large sample asymptotics for the two-parameter Poisson-Dirichlet process.” In Clarke, B. and Ghosal, S. (eds.), Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh, volume 3, 187–199. Cambridge, MA: MIT Press.
  • James, L. F., Lijoi, A., and Prünster, I. (2006). “Conjugacy as a distinctive feature of the Dirichlet process.” Scandinavian Journal of Statistics, 33: 105–120.
  • — (2009). “Posterior analysis for normalized random measures with independent increments.” Scandinavian Journal of Statistics, 36: 76–97.
  • Jang, G. H., Lee, J., and Lee, S. (2010). “Posterior consistency of species sampling priors.” Statistica Sinica, 20: 581–593.
  • Kim, N. and Bickel, P. (2003). “The limit distribution of a test statistic for bivariate normality.” Statistica Sinica, 13: 327–349.
  • Lijoi, A., Mena, R. H., and Prünster, I. (2005a). “Bayesian nonparametric analysis for a generalized Dirichlet process prior.” Statistical Inference for Stochastic Processes, 8: 283–309.
  • — (2005b). “Hierarchical mixture modelling with normalized inverse-Gaussian priors.” Journal of the American Statistical Association, 100: 1278–1291.
  • — (2007). “Controlling the reinforcement in Bayesian nonparametric mixture models.” Journal of the Royal Statistical Society: Series B, 69: 715–740.
  • Lo, A. Y. (1987). “A large sample study of the Bayesian bootstrap.” The Annals of Statistics, 15: 360–275.
  • Nieto-Barajas, L. E. and Prünster, I. (2009). “A sensitivity analysis for Bayesian nonparametric density estimators.” Statistica Sinica, 19: 685–705.
  • Pollard, D. (1984). Convergence of Stochastic Processes. New York: Springer-Verlag.
  • Shorack, G. and Wellner, J. (1986). Empirical Processes with Applications to Statistics. New York: Wiley.
  • van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes with Applications to Statistics. New York: Springer.
  • Zarepour, M. and Al Labadi, L. (2012). “On a Rapid Simulation of the Dirichlet Process.” Statistics and Probability Letters, 82: 916–924.