A likelihood for correlated extreme series

Abstract

This paper develops a likelihood for sequences of extremes when observations are dependent in time. The likelihood allows researchers to obtain more realistic standard errors of the generalized extreme‐value parameters. As a motivating example, annual minimum temperatures are examined from the Faraday/Vernadsky research station in Antarctica. Here, the year‐to‐year correlation in the series is about 0.46. Our likelihood allows the series to have temporal correlation but also keeps a generalized extreme‐value marginal distribution at each point in time. An analysis of the Faraday/Vernadsky annual minimum temperatures is conducted. While the standard error of the estimated trend increases when dependence is taken into account, it does not change the correlation‐ignored inference that annual minimum temperatures at the station are increasing.

Modeling short‐ranged dependence in block extrema with application to polar temperature data

“…Zhu et al (2019) analyze the same Antarctic data set that we consider here and conclude that the series of annual minima exhibits short‐ranged dependence, primarily at lag 1. However, their approach is based on using a Gaussian copula (Joe, 1997; Nelsen, 2006) to model dependence.…”

“…The MA(2) process that we consider is defined by for and where W is iid Gaussian white noise with unit variance. The resulting series is transformed to have GEV(0,1,−0.1) marginal distributions via probability integral transformations, similar to Zhu et al (2019). We note that this MA(2) process has a true correlation of approximately 0.42 at lag one, and a true correlation of approximately 0.06 at lag 2.…”

“…We first present an analysis of an annual minimum temperature data set at the Faraday/Vernadsky station in Antarctica (65.25°S, 64.26°W). This same time series is analyzed in Zhu et al (2019), and includes data from 1947 to 1993 (Jones & Reid, 2001); we plot the corresponding values in the left panel of Figure 2. Based on these data, in the right panel of Figure 2 we plot the estimated value of , using the estimator in Equation (19) for k = 1, … , 5.…”

“…Inference is typically performed by assuming that a series of block maxima are independent GEV realizations. In many applications, this assumption of independence among block maxima is quite reasonable (e.g., Huang et al, 2016), given that the block size is sufficiently large and the (within block) serial dependence is relatively weak; however, there are cases where dependence between consecutive block maxima is exhibited (e.g., Zhu, Liu, & Lund, 2019). In these instances, traditional block maxima analysis will ignore this dependence, potentially disregarding important information and leading to invalid inference.…”

Statistics for climate informatics

“…Huang, Nychka, and Zhang (2018) propose a new method for estimating precipitation extremes using log-histosplines-an alternative to the widely applied peaks-over-thresholds (POT) method. Zhu, Liu, and Lund (2018) propose a new approach to modeling correlated extreme time series while keeping a marginal generalized extreme value (GEV) distribution at all time points.…”