Bayesian Inference for Mixtures of Stable Distributions

In many different fields such as hydrology, telecommunications, physics of condensed matter and finance, the gaussian model results unsatisfactory and reveals difficulties in fitting data with skewness, heavy tails and multimodality. The use of stable distributions allows for modelling skewness and heavy tails but gives rise to inferential problems related to the estimation of the stable distributions' parameters. Some recent works have proposed characteristic function based estimation method and MCMC simulation based estimation techniques like the MCMC-EM method and the Gibbs sampling method in a full Bayesian approach. The aim of this work is to generalise the stable distribution framework by introducing a model that accounts also for multimodality. In particular we introduce a stable mixture model and a suitable reparametrisation of the mixture, which allow us to make inference on the mixture parameters. We use a full Bayesian approach and MCMC simulation techniques for the estimation of the posterior distribution. Finally we propose some applications of stable mixtures to financial data.

[1]  Thomas Mikosch,et al.  Modeling Dependence and Tails of Financial Time Series , 2003 .

[2]  E. Fama.  The Behavior of Stock-Market Prices , 1965 .

[3]  C. Robert.  Mixtures of Distributions: Inference and Estimation , 1996 .

[4]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[5]  Nalini Ravishanker,et al.  Bayesian inference for time series with infinite variance stable innovations , 1998 .

[6]  E. Fama.  Portfolio Analysis in a Stable Paretian Market , 1965 .

[7]  Roberto Casarin,et al.  Bayesian Inference for Mixture of Stable Distributions , 2003 .

[8]  C. Robert,et al.  Computational and Inferential Difficulties with Mixture Posterior Distributions , 2000 .

[9]  S. Richardson,et al.  Mixtures of distributions: inference and estimation , 1995 .

[10]  John Teichmoeller.  A Note on the Distribution of Stock Price Changes , 1971 .

[11]  Bruce D. Fielitz,et al.  Stable Distributions and the Mixtures of Distributions Hypotheses for Common Stock Returns , 1983 .

[12]  S. Godsill.  MCMC and EM-based methods for inference in heavy-tailed processes with /spl alpha/-stable innovations , 1999, Proceedings of the IEEE Signal Processing Workshop on Higher-Order Statistics. SPW-HOS '99.

[13]  Hoon Kim,et al.  Monte Carlo Statistical Methods , 2000, Technometrics.

[14]  A. Barnea,et al.  A Reexamination of the Empirical Distribution of Stock Price Changes , 1973 .

[15]  M. Stephens.  Bayesian analysis of mixture models with an unknown number of components- an alternative to reversible jump methods , 2000 .

[16]  竹中 茂夫.  G.Samorodnitsky,M.S.Taqqu:Stable non-Gaussian Random Processes--Stochastic Models with Infinite Variance , 1996 .

[17]  W. DuMouchel.  On the Asymptotic Normality of the Maximum-Likelihood Estimate when Sampling from a Stable Distribution , 1973 .

[18]  D. Dey,et al.  Multivariate Survival Analysis with Positive Stable Frailties , 1999, Biometrics.

[19]  Ronald L. Wasserstein,et al.  Monte Carlo: Concepts, Algorithms, and Applications , 1997 .

[20]  D. Applebaum.  Stable non-Gaussian random processes , 1995, The Mathematical Gazette.

[21]  B. Mandelbrot.  The Variation of Certain Speculative Prices , 1963 .

[22]  C. Robert,et al.  Estimation of Finite Mixture Distributions Through Bayesian Sampling , 1994 .

[23]  F. A. Seiler,et al.  Numerical Recipes in C: The Art of Scientific Computing , 1989 .

[24]  D. Buckle.  Bayesian Inference for Stable Distributions , 1995 .

[25]  Svetlozar T. Rachev,et al.  Stable Paretian modeling in finance: some empirical and theoretical aspects , 1998 .

[26]  L. Tierney.  Markov Chains for Exploring Posterior Distributions , 1994 .

[27]  C. Mallows,et al.  A Method for Simulating Stable Random Variables , 1976 .

[28]  M. Taqqu,et al.  Financial Risk and Heavy Tails , 2003 .

[29]  P. Green,et al.  On Bayesian Analysis of Mixtures with an Unknown Number of Components (with discussion) , 1997 .

[30]  Sylvia Richardson,et al.  Markov Chain Monte Carlo in Practice , 1997 .

[31]  Radford M. Neal.  Bayesian Mixture Modeling by Monte Carlo Simulation , 1991 .

[32]  Geoffrey J. McLachlan,et al.  Finite Mixture Models , 2019, Annual Review of Statistics and Its Application.

[33]  Murad S. Taqqu,et al.  A Practical Guide to Heavy Tails: Statistical Techniques for Analysing Heavy-Tailed Distributions , 1998 .

[34]  W. L. Beedles,et al.  Asymmetric Stable Distributed Security Returns , 1980 .

[35]  Richard L. Tweedie,et al.  Markov Chains and Stochastic Stability , 1993, Communications and Control Engineering Series.

[36]  M. Escobar,et al.  Bayesian Density Estimation and Inference Using Mixtures , 1995 .

[37]  Christian P. Robert,et al.  The Bayesian choice , 1994 .

[38]  E. Fama,et al.  Parameter Estimates for Symmetric Stable Distributions , 1971 .

[39]  M. Brenner.  On the Stability of the Distribution of the Market Component in Stock Price Changes , 1974, Journal of Financial and Quantitative Analysis.

[40]  Luc Devroye,et al.  Chapter 4 Nonuniform Random Variate Generation , 2006, Simulation.

[41]  P. Saama.  MAXIMUM LIKELIHOOD AND BAYESIAN METHODS FOR MIXTURES OF NORMAL DISTRIBUTIONS , 1997 .

[42]  R. Weron.  Correction to: "On the Chambers–Mallows–Stuck Method for Simulating Skewed Stable Random Variables" , 1996 .

[43]  W. Linde.  STABLE NON‐GAUSSIAN RANDOM PROCESSES: STOCHASTIC MODELS WITH INFINITE VARIANCE , 1996 .

[44]  Mark A. McComb.  A Practical Guide to Heavy Tails , 2000, Technometrics.

[45]  E. Fama,et al.  Some Properties of Symmetric Stable Distributions , 1968 .

[46]  S. Rachev,et al.  Stable Paretian Models in Finance , 2000 .