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Unformatted text preview: Bayesian Inference for Mixtures of Stable Distributions Roberto Casarin CEREMADE University Paris IX (Dauphine) and Dept. of Economics University Ca Foscari, Venice Abstract 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. Keywords : Mixture model, Stable distributions, Bayesian inference, Gibbs sampling. 1 Introduction In many different fields such as hydrology, telecommunications, physics and finance, Gaussian models reveal difficulties in fitting data that exhibits a high degree of heterogeneity; thus stable distributions have been introduced as a generalisation of the Gaussian model. Stable distributions Im thankfull to professors C.P. Robert and M. Billio for the suggestions to my Phd. thesis. This work has been presented at the Young Statistician Meeting, Cambridge 14-15 April 2003. Adress: CEREMADE, University Paris IX (Dauphine), 16 place du Mar echal de Lattre de Tassigny, 75775 Paris C edex 16. E-mail: email@example.com 1 allow also for infinite variance, skewness and heavy tails. The tails of a stable distribution decay like a power function, allowing extreme events to have higher probability mass than in Gaussian model. For a summary of the properties of the stable distributions see Zoloratev  and Samorodnitsky and Taqqu , which provide a good theoretical background on heavy- tailed distributions. The practical use of heavy-tailed distributions in many different fields is well documented in the book of Adler, Feldman and Taqqu , which also reviews the estimation techniques. In finance, the first studies on the hypothesis of stable distributed stock prices can be attributed to Mandelbrot , Fama ,  and Fama and Roll , . They propose stable distributions and give some statistical instruments for the inference on the characteristic exponent. The use of stable distributions has been motivated also on the basis of empirical evidence from financial markets. Brenner  uses the notion of stationarity of the time series toevidence from financial markets....
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