A semiparametric Bayesian approach to extreme value estimation

This paper is concerned with extreme value density estimation. The generalized Pareto distribution (GPD) beyond a given threshold is combined with a nonparametric estimation approach below the threshold. This semiparametric setup is shown to generalize a few existing approaches and enables density estimation over the complete sample space. Estimation is performed via the Bayesian paradigm, which helps identify model components. Estimation of all model parameters, including the threshold and higher quantiles, and prediction for future observations is provided. Simulation studies suggest a few useful guidelines to evaluate the relevance of the proposed procedures. They also provide empirical evidence about the improvement of the proposed methodology over existing approaches. Models are then applied to environmental data sets. The paper is concluded with a few directions for future work.

[1]  Lynn Kuo,et al.  A Bayesian predictive approach to determining the number of components in a mixture distribution , 1995 .

[2]  Dani Gamerman,et al.  Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference , 1997 .

[3]  George G. Lorentz,et al.  Constructive Approximation , 1993, Grundlehren der mathematischen Wissenschaften.

[4]  Richard L. Smith,et al.  Models for exceedances over high thresholds , 1990 .

[5]  Sérgio Luís Ganhão Vicente.  Extreme value theory: an application to sports , 2012 .

[6]  R. Fisher,et al.  Limiting forms of the frequency distribution of the largest or smallest member of a sample , 1928, Mathematical Proceedings of the Cambridge Philosophical Society.

[7]  W. J. Hall,et al.  Approximating Priors by Mixtures of Natural Conjugate Priors , 1983 .

[8]  A. O'Hagan,et al.  Accounting for threshold uncertainty in extreme value estimation , 2006 .

[9]  Richard L. Smith.  Threshold Methods for Sample Extremes , 1984 .

[10]  Dani Gamerman,et al.  A default Bayesian approach for regression on extremes , 2011 .

[11]  Bradley P. Carlin,et al.  Bayesian measures of model complexity and fit , 2002 .

[12]  Jonathan A. Tawn,et al.  A Bayesian Analysis of Extreme Rainfall Data , 1996 .

[13]  Robert B. Gramacy,et al.  Ja n 20 08 Bayesian Treed Gaussian Process Models with an Application to Computer Modeling , 2009 .

[14]  S. Cabras,et al.  A default Bayesian procedure for the generalized Pareto distribution , 2007 .

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

[16]  S. Frühwirth-Schnatter.  Markov chain Monte Carlo Estimation of Classical and Dynamic Switching and Mixture Models , 2001 .

[17]  A. W. Kemp,et al.  Applied Probability and Queues , 1989 .

[18]  Richard L. Smith.  Maximum likelihood estimation in a class of nonregular cases , 1985 .

[19]  C. Klüppelberg,et al.  Modelling Extremal Events , 1997 .

[20]  J. Pickands.  Statistical Inference Using Extreme Order Statistics , 1975 .

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

[22]  J. N. Corcoran.  Modelling Extremal Events for Insurance and Finance:Modelling Extremal Events for Insurance and Finance , 2002 .

[23]  C. Cunnane,et al.  A note on the Poisson assumption in partial duration series models , 1979 .

[24]  J. Corcoran.  Modelling Extremal Events for Insurance and Finance , 2002 .

[25]  K. F. Turkman,et al.  A Predictive Approach to Tail Probability Estimation , 2001 .

[26]  Dani Gamerman,et al.  Regression models for exceedance data via the full likelihood , 2011, Environmental and Ecological Statistics.

[27]  Mhamed-Ali El-Aroui,et al.  Quasi-Conjugate Bayes Estimates for GPD Parameters and Application to Heavy Tails Modelling , 2005, 1103.6216.

[28]  S. Asmussen,et al.  Applied Probability and Queues , 1989 .

[29]  F. Nascimento,et al.  Generalized Pareto models with time-varying tail behavior ∗ , 2010 .

[30]  L. Wasserman,et al.  Practical Bayesian Density Estimation Using Mixtures of Normals , 1997 .

[31]  A. F. Smith,et al.  Statistical analysis of finite mixture distributions , 1986 .

[32]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[33]  Michael P. Wiper,et al.  Mixtures of Gamma Distributions With Applications , 2001 .

[34]  A. F. Smith,et al.  Statistical analysis of finite mixture distributions , 1986 .

[35]  A. Frigessi,et al.  A Dynamic Mixture Model for Unsupervised Tail Estimation without Threshold Selection , 2002 .

[36]  Gareth O. Roberts,et al.  Examples of Adaptive MCMC , 2009 .

[37]  D. Gamerman,et al.  Bayesian analysis of extreme events with threshold estimation , 2004 .

[38]  S. Ahmed.  Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference (2nd ed.), by Dani Gamerman and Hedibert F. Lopes , 2008 .

[39]  A. Jenkinson.  The frequency distribution of the annual maximum (or minimum) values of meteorological elements , 1955 .