Abstract
There are many parameters in multivariate maxima of moving maxima processes—or M4 processes. However, the more parameters there are, the more difficult it is to estimate them. It is not just an issue of numerical stability, of course. The statistical precision of the estimates will be poor if the number of parameters is too large. We consider asymmetric geometric structures which correspond to special moving patterns of extreme observations in observed time series. We study the model identifiability and propose parameter estimators. All proposed estimators are shown to be consistent and asymptotically joint normal. Simulation study and real data modeling of North Sea wave height data are illustrated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Billingsley P. (1995). Probability and measure (3rd ed). Wiley, New York
Coles S.G., Tawn J.A. (1991). Modeling extreme multivariate events. Journal of Royal Statistical Society, Series B 53:377–392
Coles S.G., Tawn J.A. (1994). Statistical methods for multivariate extrems: an application to structural design (with discussion). Applied Statistics 43(1):1–48
Davis R.A., Resnick S.I. (1989). Basic properties and prediction of Max-ARMA processes. Advances in Applied Probability 21:781–803
Davis R.A., Resnick S.I. (1993). Prediction of stationary Max-stable processes. Annals of Applied Probability 3(2):497–525
Deheuvels P. (1983). Point processes and multivariate extreme values. Journal of Multivariate Analysis 13:257–272
Embrechts P., Klüppelberg C., Mikosch T. (1997). Modelling extremal events for insurance and finance. Springer, Berlin Heidelberg New York
Galambos J. (1987). Asymptotic theory of extreme order statistics (2nd ed). Krieger, Malabar
de Haan L. (1984). A Spectral Representation for Max-stable processes. Annals of Probability 12(4):1194–1204
de Haan, L. (1985). Extremes in higher dimensions: the model and some statistics. In Proceedings of 45th session international statistics institute, (paper 26.3). The Hague: International Statistical Institute.
de Haan L., Resnick S.I. (1977). Limit theory for multivariate sample extremes. Z. Wahrscheinlichkeitstheorie verw Gebiete 40:317–337
de Haan L., de Ronde J. (1998). Sea and wind: multivariate extremes at work. Extremes 1(1):7–45
Hall P., Peng L., Yao Q. (2002). Moving-maximum models for extrema of time series. Journal of Statistical Planning and Inference 103:51–63
Leadbetter M.R., Lindgren G., Rootzén H. (1983). Extremes and related properties of random sequences and processes. Springer, Berlin Heidelberg New York
Pickands J. III (1975). Statistical inference using extreme order statistics. The Annals of Statistics 3(1):119–131
Pickands, J. III (1981). Multivariate Extreme Value distributions. In Proceedings of 43rd session international statistics institute, pp. (859–878). Buenos Aires.
Resnick S.I. (1987). Extreme values, regular variation, and point processes. Springer, Berlin Heidelberg New York
Smith R.L. (1990). Extreme value theory. In: Ledermann W. (eds) Handbook of applicable mathematics (Supplement). Wiley, Chichester
Smith, R. L. (2003). Statistics of extremes, with applications in the environment, insurance and finance. In B. Finkenstadt, H. Rootzén, (Eds.) Extreme value in finance, telecommunications and the environment, to be published by Chapman and Hall/CRC.
Smith, R. L., Weissman, I. (1996). Characterization and estimation of the multivariate extremal index. Manuscript, UNC.
Van Gelder, P. H. A. J. M., de Ronde, J. G., Neykov, N. M., Neytchev, P. (2000). Regional frequency analysis of extreme wave heights: Trading space for time. In Proceedings of the 27th ICCE (pp. 1099-1112, Vol. 2,) Coastal Engineering 2000, Sydney.
Zhang, Z. (2002). Multivariate extremes, Max-stable process estimation and dynamic financial modeling. PhD Dissertation, Department of Statistics, University of North Carolina.
Zhang Z. (2005). A new class of tail-dependent time series models and its applications in financial time series. Advances in Econometrics 20(B):323–358
Zhang, Z., Smith, R. L. (2001). Modeling financial time series data as moving maxima processes. Technical Report, Department of Statistics, University of North Carolina. Submitted to 2001 NBER/NSF time series conference.
Zhang, Z., Smith, R. L. (2002). On the estimation and application of Max-stable processes. Manuscript, Washington University.
Zhang Z., Smith R.L. (2004). The behavior of multivariate maxima of moving maxima processes. Journal of Applied Probability 41:1113–1123
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Zhang, Z. The estimation of M4 processes with geometric moving patterns. AISM 60, 121–150 (2008). https://doi.org/10.1007/s10463-006-0078-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-006-0078-0