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Conditioning exceedances on covariate processes. (English) Zbl 1329.62225

Summary: This article concerns the statistical inference for the upper tail of the conditional distribution of a response variable \(Y\) given a covariate \(X=x\) based on \(n\) random vectors within the parametric extreme value framework. Pioneering work in this field was done by R. L. Smith [Stat. Sci. 4, No. 4, 367–393 (1989; Zbl 0955.62646)] and R. L. Smith and T. S. Shively [“Point process approach to modeling trends in tropospheric ozone based on exceedances of a high threshold”, Atmospheric Environment 29, No. 23, 3489–3499 (1995; doi:10.1016/1352-2310(95)00030-3)]. We propose to base the inference on a conditional distribution of the point process of exceedances given the point process of covariates. It is of importance that the conditional distribution merely depends on the conditional distribution of the response variable given the covariates. In the special case of Poisson processes such a result may be found in [R. D. Reiss, A course on point processes. New York: Springer-Verlag (1993; Zbl 0771.60037)]. Our results are valid within the broader model where the response variables are conditionally independent given the covariates. It is numerically exemplified that the maximum likelihood principle leads to more accurate estimators within the conditional approach than in the previous one.

MSC:

62G32 Statistics of extreme values; tail inference
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
62F10 Point estimation
60G70 Extreme value theory; extremal stochastic processes

Software:

ismev; FinTS
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