Abstract
The study of extreme values is of crucial interest in many contexts. The concentration of pollutants, the sea-level and the closing prices of stock indexes are only a few examples in which the occurrence of extreme values may lead to important consequences. In the present paper we are interested in detecting trend in sample extremes. A common statistical approach used to identify trend in extremes is based on the generalized extreme value distribution, which constitutes a building block for parametric models. However, semiparametric procedures imply several advantages when exploring data and checking the model. This paper outlines a semiparametric approach for smoothing sample extremes, based on nonlinear dynamic modelling of the generalized extreme value distribution. The relative merits of this approach are illustrated through two real examples.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anderson, B.D.O. and Moore, J.B., Optimal Filtering, Prentice-Hall, Englewood Cliffs, 1979.
Chavez-Demoulin, V., Two Problems in Environmental Statistics: Capture-recapture Analysis and Smooth Extremal Models, PhD Thesis. Department of Mathematics, Swiss Federal Institute of Technology, Lausanne, 1999.
Coles, S., An Introduction to Statistical Modeling of Extreme Values, Springer, London, 2001.
Coles, S.G. and Tawn, J.A., “A Bayesian analysis of extreme rainfall data,” Appl. Stat. 45, 463–478, (1996).
Conover, W.J., Practical Nonparametric Statistics, Wiley, New York, 1999.
Davison, A.C. and Ramesh, N.I., “Local likelihood smoothing of sample extremes,” J. R. Stat. Soc., B 62, 191–208, (2000).
De Jong, P. and Mazzi, S., “Modeling and smoothing unequally spaced sequence data,” Stat. Inference Stoch. Process. 4, 53–71, (2001).
Doucet, A., De Freitas, N., and Gordon, N., Sequential Monte Carlo Methods in Practice, Springer, New York, 2001.
Durbin, J. and Koopman, S.J., Time Series Analysis by State Space Methods, Oxford University Press, Oxford, 2001.
Easterling, D.R., Meehl, G.A., Parmesan, C., Changnon, S.A., Karl, T.R., and Mearns, L.O., “Climate extremes: Observations, modeling, and impacts,” Science 289, 2068–2074, (2000).
Fahrmeir, L. and Knorr-Held, L., “Dynamic and semiparametric models,” in Smoothing and Regression: Approaches, Computation and Application (M. Schimek, ed) Wiley, New York, 513–544, (2000).
Gelman, A., Meng, X.L. and Stern, H., “Posterior predictive assessment of model fitness via realized discrepancies,” Stat. Sin. 6, 733–807, (1996).
Gordon, N.J., Salmond, D.J., and Smith, A.F.M., “Novel approach to nonlinear/non-Gaussian Bayesian state estimation,” IEE Proc. F 140, 107–113, (1993).
Guttman, I., “The use of the concept of a future observation in goodness-of-fit problems,” J. R. Stat. Soc., B 29, 83–100, (1967).
Hall, P. and Tajvidi, N., “Nonparametric analysis of temporal trend when fitting parametric models to extreme-value data,” Stat. Sci. 18, 153–167, (2000).
Harvey, A.C., Forecasting, Structural Time Series Models and the Kalman Filter, Cambridge University Press, Cambridge, 1989.
Hüsler, J., “Extreme values of non-stationary random sequences,” J. Appl. Probab. 23, 937–950, (1986).
Katz, R.W., Parlange, M.B., and Naveau, P., “Statistics of extremes in hydrology,” Adv. Water Resour. 25, 1287–1304, (2002).
Kitagawa, G., “Monte-Carlo filter and smoother for non-Gaussian nonlinear state-space models,” J. Comput. Graph. Stat. 5, 1–25, (1996).
Kitagawa, G., “A self-organizing state-space model,” J. Am. Stat. Assoc. 93, 1203–1215, (1998).
Knorr-Held, L., “Conditional prior proposals in dynamic models,” Scand. J. Statist. 26, 129–144, (1999).
Kohn, R. and Ansley, C.F., “A fast algorithm for signal extraction, influence and cross-validation in state space models,” Biometrika 76, 65–79, (1989)
Leadbetter, K.R., Lindgren, G., and Rootzén, H., Extremes and Related Properties of Random Sequences and Series, Springer, New York, (1983).
Mann, M.E., Park, J., and Bradley, R.S., “Global interdecadal and century-scale climate oscillations during the past five centuries,” Nature 378, 266–270, (1995).
Pauli, F. and Coles, S.G., “Penalized likelihood inference in extreme value analyses,” J. Appl. Stat. 28, 547–560, (2001).
Piegorsch, W.W., Smith, E.P., Edwards, D. and Smith, R.L., “Statistical advances in environmental science,” Stat. Sci. 13, 186–208, (1998).
Robert, C.P. and Casella, G., Monte Carlo Statistical Methods, Springer, New York, 1999.
Robinson, M.E. and Tawn, J.A., “Statistics for exceptional athletics records,” Appl. Stat. 44, 499–511, (1995).
Robinson, M.E. and Tawn, J.A., “Reply to comment on ‘statistics for exceptional athletics records’ (by R. L. Smith),” Appl. Stat. 46, 127–128, (1997).
Rubin, D.B., “Bayesianly justifiable and relevant frequency calculations for the applied statistician,” Ann. Stat. 12, 1151–1172, (1984).
Smith, R.L., “A survey of nonregular problems,” Bull. Internat. Statist. Inst. 53, 353–372, (1989).
Smith, R.L., “Comment on ‘statistics for exceptional athletics records’ (by M. E. Robinson and J. A. Tawn),” Appl. Stat. 46, 123–127, (1997).
Smith, R.L. and Miller, J.E., “A non-Gaussian state space model and application to prediction of records,” J. R. Stat. Soc., B 48, 79–88, (1986).
Tanizaki, H., “Nonlinear and non-Gaussian state-space modeling with Monte Carlo techniques: a survey and comparative study,” in Handbook of Statistics (Stochastic Processes: Modeling and Simulation) (C.R. Rao and D.N. Shanbhag, eds) North Holland, Amsterdam, 871–929, 2003.
Wahba, G., “Improper priors, spline smoothing and the problem of guarding against model errors in regression,” J. R. Stat. Soc., B 40, 364–372, (1978).
Wecker, W.E. and Ansley, C.F., “The signal extraction approach to nonlinear regression and spline smoothing,” J. Am. Stat. Assoc. 78, 81–89, (1983).
Author information
Authors and Affiliations
Corresponding author
Additional information
AMS 2000 Subject Classification. Primary—62G32, 62G05, 62M10
Rights and permissions
About this article
Cite this article
Gaetan, C., Grigoletto, M. Smoothing Sample Extremes with Dynamic Models. Extremes 7, 221–236 (2004). https://doi.org/10.1007/s10687-005-6474-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10687-005-6474-7