Robust Estimation for the Weibull Process Applied to Eruption Records
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The Weibull process is a parsimoniously parameterized nonhomogeneous Poisson process with monotonic trend, which has been widely used in reliability applications. It has also been used in volcanology to model the process of eruption onsets for a volcano with waning or waxing activity, and thus produce hazard forecasts. However, particularly in the latter application, problems with missing or spurious data can strongly influence the parameter estimates, which are usually obtained by maximizing the log likelihood function, and hence the future hazard. We show how theory developed for robust estimation of a nonhomogeneous Poisson process can be implemented for the Weibull process. The flank eruptions of Mt. Etna, in Sicily, is one of the most complete and best studied records of volcanism. Nevertheless, a number of different catalogs exist. We show how these can be at least partially reconciled by robust estimation, and how the more dubious regions of the catalogs can be identified.
KeywordsFlank eruptions M-estimator Mt. Etna Nonhomogeneous Poisson process
This work was carried out while the first author was supported as a Massey University postdoctoral fellow by the Natural Hazards Research Platform and the Earthquake Commission. We thank an anonymous reviewer and the editor for helpful suggestions on the original manuscript.
- Behncke B, Neri M, Nagay A (2005) Lava flow hazard at Mount Etna (Italy): new data from a GIS-based study. Special Paper 396, Geological Society of America Google Scholar
- Branca S, Del Carlo P (2004) Eruptions of Mt. Etna during the past 3,200 years: a revised compilation integrating the historical and stratigraphic records. In: Bonaccorso A, Calvari S, Coltelli M, Del Negro C, Falsaperla S (eds) Mt. Etna: volcano laboratory. Geophysical monograph series, vol 143. Am Geophys Union, Washington, pp 1–27 CrossRefGoogle Scholar
- Crow TH (1974) Reliability analysis for complex repairable systems. In: Proschan F, Serfling RJ (eds) Reliability and biometry. SIAM, Philadelphia, pp 379–410 Google Scholar
- Daley DJ, Vere-Jones D (1988) An introduction to the theory of point processes. Springer, New York Google Scholar
- Siebert L, Simkin T (2002) Volcanoes of the world: an illustrated catalog of Holocene Volcanoes and their eruptions. http://www.volcano.si.edu/world/
- Stefanski LA, Carroll RJ, Ruppert D (1986) Optimally bounded score functions for generalized linear-models with applications to logistic regression. Biometrika 73:413–424 Google Scholar
- Vere-Jones D (1992) Statistical methods for the description and display of earthquake catalogs. In: Walden AT, Guttorp P (eds) Statistics in the environmental and earth sciences. Edward Arnold, London, pp 220–246 Google Scholar