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Estimation of the upper cutoff parameter for the tapered Pareto distribution

Part of: Parametric inference

Published online by Cambridge University Press:  14 July 2016

Y. Y. Kagan  and
F. Schoenberg
Y. Y. Kagan*
Affiliation:
University of California
F. Schoenberg*
Affiliation:
University of California
*
1Postal address: Department of Earth and Space Sciences, University of California, Los Angeles, CA 90095–1567, USA. Email: [email protected]
2Postal address: Department of Statistics, University of California, Los Angeles, CA 90095–1554, USA. Email: [email protected]
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Abstract

The tapered (or generalized) Pareto distribution, also called the modified Gutenberg-Richter law, has been used to model the sizes of earthquakes. Unfortunately, maximum likelihood estimates of the cutoff parameter are substantially biased. Alternative estimates for the cutoff parameter are presented, and their properties discussed.

Keywords

Parameter estimationtapered Pareto distributionGutenberg-Richter relationmaximum likelihood estimationmethod of momentsearthquakes

MSC classification

Primary: 62F10: Point estimation
Secondary: 62F12: Asymptotic properties of estimators 62F25: Tolerance and confidence regions 62F15: Bayesian inference
Type
Models and statistics in seismology
Information
Journal of Applied Probability , Volume 38 , Issue A: Probability, Statistics and Seismology , 2001 , pp. 158 - 175
Copyright
Copyright © Applied Probability Trust 2001 

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References

[1] Bird, P., Kagan, Y. Y. and Jackson, D. D. (2000). Frequency-magnitude distribution, effective lithosphere thickness, and seismic efficiency of oceanic transforms and spreading ridges (abstract). Eos Trans. AGU 81(22), p. WP147.Google Scholar
[2] Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
[3] David, F. N. and Johnson, N. L. (1951). The effect of non-normality on the power function of the F-test in the analysis of variance. Biometrika 38, 43–57.CrossRefGoogle ScholarPubMed
[4] Dusenberry, W. E. and Bowman, K. O. (1977). The moment estimator for the shape parameter of the Gamma distribution. Comm. Stat. B 1, 1–19.Google Scholar
[5] Dziewonski, A. M., Ekström, G. and Maternovskaya, N. N. (2000). Centroid-moment tensor solutions for July-September 1999. Phys. Earth Planet. Inter. 119, 311–319.Google Scholar
[6] Frohlich, C. and Davis, S. D. (1999). How well constrained are well-constrained T, B, and P axes in moment tensor catalogs? J. Geophys. Res. 104, 4901–4910.Google Scholar
[7] Jackson, D. D. and Kagan, Y. Y. (1999). Testable earthquake forecasts for 1999. Seismol. Res. Lett. 70, 393–403.Google Scholar
[8] Jenkins, G. M. and Watts, D. G. (1968). Spectral Analysis and its Applications. Holden-Day, San Francisco.Google Scholar
[9] Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995). Continuous Univariate Distributions , 2nd edn. Wiley, New York.Google Scholar
[10] Kagan, Y. Y. (1993). Statistics of characteristic earthquakes. Bull. Seismol. Soc. Amer. 83, 7–24.Google Scholar
[11] Kagan, Y. Y. (1994). Observational evidence for earthquakes as a nonlinear dynamic process. Physica D 77, 160–192.Google Scholar
[12] Kagan, Y. Y. (1999). Universality of the seismic moment-frequency relation. Pure Appl. Geophys. 155, 537–573.CrossRefGoogle Scholar
[13] Kagan, Y. Y. and Knopoff, L. (1984). A stochastic model of earthquake occurrence. Proc. 8th Int. Conf. Earthq. Engrg. , San Francisco, CA, Vol. 1, 295–302.Google Scholar
[14] Kagan, Y. Y. and Vere-Jones, D. (1996). Problems in the modelling and statistical analysis of earthquakes. In Athens Conference on Applied Probability and Time Series Analysis (Lecture Notes Statist. 114), eds Heyde, C. C., Prohorov, Yu. V., Pyke, R. and Rachev, S. T., Springer, New York, 398–425.Google Scholar
[15] Knopoff, L. and Kagan, Y. Y. (1977). Analysis of the theory of extremes as applied to earthquake problems. J. Geophys. Res. 82, 5647–5657.Google Scholar
[16] Lay, T. and Wallace, T. C. (1995). Modern Global Seismology. Academic Press, San Diego.Google Scholar
[17] Main, I. G. (1996). Statistical physics, seismogenesis, and seismic hazard. Rev. Geophys. 34, 433–462.CrossRefGoogle Scholar
[18] Molchan, G. M. and Podgaetskaya, V. M. (1973). Parameters of global seismicity, I. In Computational Seismology 6, ed. Keilis-Borok, V. I., Nauka, Moscow, 44–66 (in Russian).Google Scholar
[19] Ogata, Y. and Katsura, K. (1993). Analysis of temporal and spatial heterogeneity of magnitude frequency distribution inferred from earthquake catalogues. Geophys. J. Int. 113, 727–738.Google Scholar
[20] Pareto, V. (1897). Cours d'économie Politique , Vol. 2. F. Rouge, Lausanne. (Quoted by Pareto, V. (1964), Œuvres Complètes, Vol. II, Libraire Droz, Genève.) Google Scholar
[21] Pisarenko, V. F. (1991). Statistical evaluation of maximum possible earthquakes. Phys. Solid Earth 27, 757–763 (English translation).Google Scholar
[22] Pisarenko, V. F., Lyubushin, A. A., Lysenko, V. B. and Golubeva, T. V. (1996). Statistical estimation of seismic hazard parameters: maximum possible magnitude and related parameters. Bull. Seismol. Soc. Amer. 86, 691–700.CrossRefGoogle Scholar
[23] Shenton, L. R., Bowman, K. O. and Sheehan, D. (1971). Sampling moments of moments associated with univariate distributions. J. Roy. Statist. Soc. Ser. B 33, 444–457.Google Scholar
[24] Sipkin, S. A., Bufe, C. G. and Zirbes, M. D. (2000). Moment-tensor solutions estimated using optimal filter theory: global seismicity, 1998. Phys. Earth Planet. Inter. 118, 169–179.CrossRefGoogle Scholar
[25] Sornette, D. and Sornette, A. (1999). General theory of the modified Gutenberg-Richter law for large seismic moments. Bull. Seismol. Soc. Amer. 89, 1121–1030.Google Scholar
[26] Utsu, T. (1999). Representation and analysis of the earthquake size distribution: a historical review and some new approaches. Pure Appl. Geophys. 155, 509–535.CrossRefGoogle Scholar
[27] Vere-Jones, D. (1976). A branching model for crack propagation. Pure Appl. Geophys. 114, 711–725.Google Scholar
[28] Vere-Jones, D. (1977). Statistical theories of crack propagation. Math. Geol. 9, 455–481.CrossRefGoogle Scholar
[29] Vere-Jones, D. (1992). Statistical methods for the description and display of earthquake catalogues. In Statistics in the Environmental and Earth Sciences , eds Walden, A. T. and Guttorp, P., Arnold, London, 220–244.Google Scholar
[30] Vere-Jones, D., Robinson, R. and Yang, W. Z. (2001). Remarks on the accelerated moment release model: problems of model formulation simulation and estimation. Geophys. J. Int. 144, 517–531.CrossRefGoogle Scholar
[31] Wilks, S. S. (1962). Mathematical Statistics. Wiley, New York.Google Scholar
[32] Wyss, M. (1973). Towards a physical understanding of the earthquake frequency distribution. Geophys. J. Roy. Astronom. Soc. 31, 341–359.CrossRefGoogle Scholar