Extremal clustering in non-stationary random sequences

• Published in: Extremes (Boston) Vol. 24; no. 4; pp. 725 - 752
• Main Authors: Auld, Graeme, Papastathopoulos, Ioannis
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2021; Springer Nature B.V
• Subjects: Civil Engineering; Clustering; Economics; Environmental Management; Extreme values; Finance; Hydrogeology; Insurance; Management; Mathematics and Statistics; Quality Control; Random variables; Reliability; Safety and Risk; Statistics; Statistics for Business; Clustering of extremes; Non-stationary sequences; Extremal index; Interexceedance times; 60G70; Periodic processes; Intervals estimator
• ISSN: 1386-1999; 1572-915X
• DOI: 10.1007/s10687-021-00418-2

It is well known that the distribution of extreme values of strictly stationary sequences differ from those of independent and identically distributed sequences in that extremal clustering may occur. Here we consider non-stationary but identically distributed sequences of random variables subject to suitable long range dependence restrictions. We find that the limiting distribution of appropriately normalized sample maxima depends on a parameter that measures the average extremal clustering of the sequence. Based on this new representation we derive the asymptotic distribution for the time between consecutive extreme observations and construct moment and likelihood based estimators for measures of extremal clustering. We specialize our results to random sequences with periodic dependence structure.