The Gumbel hypothesis test for left censored observations using regional earthquake records as an example

Abstract. Annual maximum (AM) time series are incomplete (i.e., censored) when no events are included above the assumed censoring threshold (i.e., magnitude of completeness). We introduce a distrtibutional hypothesis test for left-censored Gumbel observations based on the probability plot correlation coefficient (PPCC). Critical values of the PPCC hypothesis test statistic are computed from Monte-Carlo simulations and are a function of sample size, censoring level, and significance level. When applied to a global catalog of earthquake observations, the left-censored Gumbel PPCC tests are unable to reject the Gumbel hypothesis for 45 of 46 seismic regions. We apply four different field significance tests for combining individual tests into a collective hypothesis test. None of the field significance tests are able to reject the global hypothesis that AM earthquake magnitudes arise from a Gumbel distribution. Because the field significance levels are not conclusive, we also compute the likelihood that these field significance tests are unable to reject the Gumbel model when the samples arise from a more complex distributional alternative. A power study documents that the censored Gumbel PPCC test is unable to reject some important and viable Generalized Extreme Value (GEV) alternatives. Thus, we cannot rule out the possibility that the global AM earthquake time series could arise from a GEV distribution with a finite upper bound, also known as a reverse Weibull distribution. Our power study also indicates that the binomial and uniform field significance tests are substantially more powerful than the more commonly used Bonferonni and false discovery rate multiple comparison procedures.