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A Monte Carlo Markov chain algorithm for a class of mixture time series models

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Abstract

This article generalizes the Monte Carlo Markov Chain (MCMC) algorithm, based on the Gibbs weighted Chinese restaurant (gWCR) process algorithm, for a class of kernel mixture of time series models over the Dirichlet process. This class of models is an extension of Lo’s (Ann. Stat. 12:351–357, 1984) kernel mixture model for independent observations. The kernel represents a known distribution of time series conditional on past time series and both present and past latent variables. The latent variables are independent samples from a Dirichlet process, which is a random discrete (almost surely) distribution. This class of models includes an infinite mixture of autoregressive processes and an infinite mixture of generalized autoregressive conditional heteroskedasticity (GARCH) processes.

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References

  • Aldous, D.J.: Exchangeability and related topics. In: École d’Été de Probabilités de Saint-Flour XIII-1983. Lecture Notes in Mathematics, vol. 1117, pp. 1–198. Springer, Berlin (1985)

    Chapter  Google Scholar 

  • Basu, S., Chib, S.: Marginal likelihood and Bayes factors for Dirichlet process mixture models. J. Am. Stat. Assoc. 98, 224–235 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  • Beal, M.J., Ghahramani, Z., Rasmussen, C.E.: The infinite hidden Markov model. Neural Inf. Process. Syst. 14, 577–585 (2002)

    Google Scholar 

  • Blackwell, D., MacQueen, J.B.: Ferguson distribution via Pólya Urn Scheme. Ann. Stat. 1, 353–355 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  • Bollerslev, T.: Generalized autoregressive conditional heteroskedasticity. J. Econom. 31, 307–327 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  • Brunner, L.J.: Using the Gibbs Sampler to simulate from the Bayes estimate of a decreasing density. Commun. Stat.—Theory Method 24, 215–226 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  • Chen, C.W.S., So, M.K.P.: On a threshold heteroscedastic model. Int. J. Forecast. 22, 73–89 (2006)

    Article  MATH  Google Scholar 

  • Chib, S.: Marginal likelihood from the Gibbs output. J. Am. Stat. Assoc. 90, 1313–1321 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  • Dahl, D.B.: Modal clustering in a class of product partition models. Technical report (2006a). Available at http://www.stat.tamu.edu/~dahl/papers/modal/tr1085.pdf

  • Dahl, D.B.: Model-based clustering for expression data via a Dirichlet process mixture model. In: Do, K.A., Müller, P., Vannucci, M. (eds.) Bayesian Inference for Gene Expression and Proteomics, pp. 97–115. Cambridge University Press, Cambridge (2006b)

    Google Scholar 

  • Engle, R.: Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1008 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  • Escobar, M.D., West, M.: Bayesian density estimation and inference using mixtures. J. Am. Stat. Assoc. 90, 577–588 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  • Escobar, M.D., West, M.: Computing nonparametric hierarchical models. Pract. Nonparamet. Semiparametr. Bayesian Stat. 1–22 (1998)

  • Ferguson, T.S.: A Bayesian analysis of some nonparametric problems. Ann. Stat. 1, 209–230 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  • Green, P.J., Richardson, S.: Modelling heterogeneity with and without the Dirichlet process. Scand. J. Stat. 28, 355–375 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  • Ghosh, R.V., Ramamoorthi, R.V.: Bayesian Nonparametrics. Springer, Berlin (2003)

    MATH  Google Scholar 

  • Griffin, J.E., Steel, M.F.J.: Order-based dependent Dirichlet processes. J. Am. Stat. Assoc. 101, 179–194 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • Hass, M., Mittnik, S., Paolella, M.S.: Asymmetric multivariate normal mixture GARCH. Comput. Stat. Data Anal. 53, 2129–2154 (2009)

    Article  Google Scholar 

  • Ho, M.W.: A Bayes method for a monotone hazard rate via S-paths. Ann. Stat. 34, 820–836 (2006a)

    Article  MATH  Google Scholar 

  • Ho, M.W.: Bayes estimation of a symmetric unimodal density via S-paths. J. Comput. Graph. Stat. 15, 848–860 (2006b)

    Article  Google Scholar 

  • Ishwaran, H., James, L.F.: Gibbs sampling methods for stick-breaking priors. J. Am. Stat. Assoc. 96, 161–173 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  • Ishwaran, H., James, L.F.: Generalized weighted Chinese restaurant processes for species sampling mixture models. Stat. Sin. 13, 1211–1235 (2003a)

    MATH  MathSciNet  Google Scholar 

  • Ishwaran, H., James, L.F.: Some further developments for stick-breaking priors: finite and infinite clustering and classification. Sankhya Ser. A 65, 577–592 (2003b)

    MATH  MathSciNet  Google Scholar 

  • Ishwaran, H., James, L.F.: Computational methods for multiplicative intensity models using weighted gamma processes: proportional hazards, marked point processes and panel count data. J. Am. Stat. Assoc. 99, 175–190 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  • James, L.F.: Poisson process partition calculus with applications to exchangeable models and Bayesian Nonparametrics (2002). Available at http://arXiv.org/abs/math/0205093

  • James, L.F.: Bayesian Poisson process partition calculus with an application to Bayesian Lévy moving averages. Ann. Stat. 33, 1771–1799 (2005)

    Article  MATH  Google Scholar 

  • James, L.F., Lijoi, A., Prüenster, I.: Bayesian inference for classes of normalized random measures. Ann. Appl. Probab. 18, 521–511 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  • Kingman, J.F.C.: Random discrete distributions. J. R. Stat. Soc., Ser. B 37, 1–22 (1975)

    MATH  MathSciNet  Google Scholar 

  • Kingman, J.F.C.: Poisson Processes. Oxford University Press, London (1993)

    MATH  Google Scholar 

  • Lau, J.W., Green, P.J.: Bayesian model based clustering procedures. J. Comput. Graph. Stat. 16, 526–558 (2007)

    Article  MathSciNet  Google Scholar 

  • Lau, J.W., Lo, A.Y.: Model based clustering and weighted Chinese restaurant processes. In: Nair, V. (ed.) Advances in Statistical Modeling and Inference: Essays in Honor of Kjell A. Doksum, pp. 405–424. World Scientific, Singapore (2007)

    Chapter  Google Scholar 

  • Lau, J.W., Siu, T.K.: Modelling long-term investment returns via Bayesian infinite mixture time series models. Scand. Actuar. J. 108, 243–282 (2008)

    Article  MathSciNet  Google Scholar 

  • Lau, J.W., So, M.K.P.: Bayesian mixture of autoregressive models. Comput. Stat. Data Anal. 53, 38–60 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  • Le, N.D., Martin, R.D., Raftery, A.E.: Modeling flat stretches, bursts, and outliers in time series using mixture transition distribution models. J. Am. Stat. Assoc. 91, 1504–1514 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  • Lijoi, A., Mena, R.H., Prünster, I.: Hierarchical mixture modeling with normalized inverse-Gaussian priors. J. Am. Stat. Assoc. 100, 1278–1291 (2005)

    Article  MATH  Google Scholar 

  • Lijoi, A., Mena, R.H., Prünster, I.: Controlling the reinforcement in Bayesian nonparametric mixture models. J. R. Stat. Soc. Ser. B 69, 715–740 (2007)

    Article  Google Scholar 

  • Lo, A.Y.: On a class of Bayesian nonparametric estimates. I. Density estimates. Ann. Stat. 12, 351–357 (1984)

    Article  MATH  Google Scholar 

  • Lo, A.Y.: Weighted Chinese restaurant processes. COSMOS 1, 59–63 (2005)

    Article  Google Scholar 

  • Lo, A.Y., Brunner, L.J., Chan, A.T.: Weighted Chinese restaurant processes and Bayesian mixture models. Research Report, Hong Kong University of Science and Technology (1996). Available at http://www.erin.utoronto.ca/~jbrunner/papers/wcr96.pdf

  • MacEachern, S.N.: Estimating normal means with a conjugate style Dirichlet process prior. Commun. Stat.—Simul. Comput. 23, 727–741 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  • MacEachern, S.N., Müller, P.: Estimating mixture of Dirichlet process models. J. Comput. Graph. Stat. 7, 223–238 (1998)

    Article  Google Scholar 

  • MacEachern, S.N., Müller, P.: Efficient MCMC schemes for robust model extensions using encompassing Dirichlet process mixture models. In: Ruggeri, F., Rios-Insua, D. (eds.) Robust Bayesian Analysis, pp. 295–316. Springer, New York (2000)

    Google Scholar 

  • Müller, P., West, M., MacEachern, S.N.: Bayesian models for non-linear auto-regressions. J. Time Ser. Anal. 18, 593–614 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  • Neal, R.M.: Markov chain sampling methods for Dirichlet process mixture models. J. Comput. Graph. Stat. 9, 249–265 (2000)

    Article  MathSciNet  Google Scholar 

  • Nelson, D.B.: Conditional heteroskedasticity in asset pricing: a new approach. Econometrica 59, 347–370 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  • Pitman, J.: Exchangeable and partially exchangeable random partitions. Probab. Theory Relat. Fields 102, 145–158 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  • Pitman, J.: Some developments of the Blackwell-MacQueen urn scheme. In: Ferguson, T.S., Shapley, L.S., MacQueen, J.B. (eds.) Statistics, Probability and Game Theory. Papers in Honor of David Blackwell. IMS Lecture Notes, Monograph Series, vol. 30, pp. 245–267 (1996)

  • Pitman, J.: Poisson Kingman partitions. In: D.R. Goldstein (ed) Science and Statistics: A Festschrift for Terry Speed. IMS Lecture Notes, Monograph Series, vol. 40, pp. 1–34 (2003)

  • Pitman, J., Yor, M.: The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25, 855–900 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  • Quintana, F.: A predictive view of Bayesian clustering. J. Stat. Plan. Inference 136, 2407–2429 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • Richardson, S., Green, P.J.: On Bayesian analysis of mixtures with an unknown number of components. J. R. Stat. Soc., Ser. B 59, 731–792 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  • West, M., Müller, P., Escobar, M.D.: Hierarchical priors and mixture models, with applications in regression and density estimation. In: Freeman, P.R., Smith, A.F.M. (eds.) A Tribute to D.V. Lindley, pp. 363–386. Wiley, New York (1994)

    Google Scholar 

  • Wong, C.S., Li, W.K.: On a mixture autoregressive model. J. R. Stat. Soc., Ser. B 62, 95–115 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • Wong, C.S., Li, W.K.: On a logistic mixture autoregressive model. Biometrika 88, 833–846 (2001a)

    Article  MATH  MathSciNet  Google Scholar 

  • Wong, C.S., Li, W.K.: On a mixture autoregressive conditional heteroscedastic model. J. Am. Stat. Assoc. 96, 982–995 (2001b)

    Article  MATH  MathSciNet  Google Scholar 

  • Zhang, Z., Li, W.K., Yuen, K.C.: On a mixture GARCH time-series model. J. Time Ser. Anal. 27, 577–597 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley 6009, Perth, Australia

    John W. Lau

  2. Department of Information Systems, Business Statistics and Operations Management, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, Hong Kong

    Mike K. P. So

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  1. John W. Lau
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  2. Mike K. P. So
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Correspondence to John W. Lau.

Additional information

The research of the first author is partly supported by Hong Kong RGC Grant #601707.

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Lau, J.W., So, M.K.P. A Monte Carlo Markov chain algorithm for a class of mixture time series models. Stat Comput 21, 69–81 (2011). https://doi.org/10.1007/s11222-009-9147-6

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  • DOI: https://doi.org/10.1007/s11222-009-9147-6

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