Uniform rates of convergence in extreme-value theory

1982 ◽  
Vol 14 (3) ◽  
pp. 600-622 ◽  
Author(s):  
Richard L. Smith
Keyword(s):  
Extreme Value Theory ◽  
Error Term ◽  
Extreme Value Distribution ◽  
Rates Of Convergence ◽  
Value Theory ◽  
Extreme Value ◽  
Value Distribution ◽  
Convergence In Distribution ◽  
Principal Error ◽  
Close Parallel

Rates of convergence are derived for the convergence in distribution of renormalised sample maxima to the appropriate extreme-value distribution. Related questions which are discussed include the estimation of the principal error term and the optimality of the renormalising constants. Throughout the paper a close parallel is drawn with the theory of slow variation with remainder. This theory is used in proving most of the results. Some applications are discussed, including some models of importance in reliability.

Uniform rates of convergence in extreme-value theory

1982 ◽  
Vol 14 (03) ◽  
pp. 600-622 ◽  
Author(s):  
Richard L. Smith
Keyword(s):  
Extreme Value Theory ◽  
Error Term ◽  
Extreme Value Distribution ◽  
Rates Of Convergence ◽  
Value Theory ◽  
Extreme Value ◽  
Value Distribution ◽  
Convergence In Distribution ◽  
Principal Error ◽  
Close Parallel

Rates of convergence are derived for the convergence in distribution of renormalised sample maxima to the appropriate extreme-value distribution. Related questions which are discussed include the estimation of the principal error term and the optimality of the renormalising constants. Throughout the paper a close parallel is drawn with the theory of slow variation with remainder. This theory is used in proving most of the results. Some applications are discussed, including some models of importance in reliability.

Convergence rates for the ultimate and pentultimate approximations in extreme-value theory

1982 ◽  
Vol 14 (04) ◽  
pp. 833-854 ◽  
Author(s):  
Jonathan P. Cohen
Keyword(s):  
Extreme Value Theory ◽  
Convergence Rates ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Rates Of Convergence ◽  
Value Theory ◽  
Extreme Value ◽  
Value Distribution ◽  
Type I ◽  
Type Ii

Let F be a distribution in the domain of attraction of the type I extreme-value distribution Λ(x). In this paper we derive uniform rates of convergence of Fn to Λfor a large class of distributions F. We also generalise the penultimate approximation of Fisher and Tippett (1928) and show that for many F a type II or type III extreme-value distribution gives a better approximation than the limiting type I distribution.

Convergence rates for the ultimate and pentultimate approximations in extreme-value theory

1982 ◽  
Vol 14 (4) ◽  
pp. 833-854 ◽  
Author(s):  
Jonathan P. Cohen
Keyword(s):  
Extreme Value Theory ◽  
Convergence Rates ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Rates Of Convergence ◽  
Value Theory ◽  
Extreme Value ◽  
Value Distribution ◽  
Type I ◽  
Type Ii

Let F be a distribution in the domain of attraction of the type I extreme-value distribution Λ(x). In this paper we derive uniform rates of convergence of Fn to Λfor a large class of distributions F. We also generalise the penultimate approximation of Fisher and Tippett (1928) and show that for many F a type II or type III extreme-value distribution gives a better approximation than the limiting type I distribution.

Extreme values of phase-type and mixed random variables with parallel-processing examples

1999 ◽  
Vol 36 (01) ◽  
pp. 194-210 ◽  
Author(s):  
Sungyeol Kang ◽  
Richard F. Serfozo
Keyword(s):  
Parallel Processing ◽  
Asymptotic Distribution ◽  
Extreme Values ◽  
Random Variables ◽  
Extreme Value Distribution ◽  
Rates Of Convergence ◽  
Value Theory ◽  
Extreme Value ◽  
Independent Random Variables ◽  
Phase Type

A basic issue in extreme value theory is the characterization of the asymptotic distribution of the maximum of a number of random variables as the number tends to infinity. We address this issue in several settings. For independent identically distributed random variables where the distribution is a mixture, we show that the convergence of their maxima is determined by one of the distributions in the mixture that has a dominant tail. We use this result to characterize the asymptotic distribution of maxima associated with mixtures of convolutions of Erlang distributions and of normal distributions. Normalizing constants and bounds on the rates of convergence are also established. The next result is that the distribution of the maxima of independent random variables with phase type distributions converges to the Gumbel extreme-value distribution. These results are applied to describe completion times for jobs consisting of the parallel-processing of tasks represented by Markovian PERT networks or task-graphs. In these contexts, which arise in manufacturing and computer systems, the job completion time is the maximum of the task times and the number of tasks is fairly large. We also consider maxima of dependent random variables for which distributions are selected by an ergodic random environment process that may depend on the variables. We show under certain conditions that their distributions may converge to one of the three classical extreme-value distributions. This applies to parallel-processing where the subtasks are selected by a Markov chain.

Statistics of Extreme Wind Speeds and Wave Heights by the Bivariate ACER Method

2015 ◽  
Vol 137 (2) ◽  
Author(s):  
Arvid Naess ◽  
Oleh Karpa
Keyword(s):  
Time Series ◽  
Extreme Value Distribution ◽  
Offshore Structures ◽  
Accurate Estimate ◽  
Value Theory ◽  
Extreme Value ◽  
Value Distribution ◽  
Extreme Value Statistics ◽  
Wind Speeds ◽  
Wave Heights

In the reliability engineering and design of offshore structures, probabilistic approaches are frequently adopted. They require the estimation of extreme quantiles of oceanographic data based on the statistical information. Due to strong correlation between such random variables as, e.g., wave heights and wind speeds (WS), application of the multivariate, or bivariate in the simplest case, extreme value theory is sometimes necessary. The paper focuses on the extension of the average conditional exceedance rate (ACER) method for prediction of extreme value statistics to the case of bivariate time series. Using the ACER method, it is possible to provide an accurate estimate of the extreme value distribution of a univariate time series. This is obtained by introducing a cascade of conditioning approximations to the true extreme value distribution. When it has been ascertained that this cascade has converged, an estimate of the extreme value distribution has been obtained. In this paper, it will be shown how the univariate ACER method can be extended in a natural way to also cover the case of bivariate data. Application of the bivariate ACER method will be demonstrated for measured coupled WS and wave height data.

Extreme rainfall from Tropical Cyclones described through the Metastatistical Extreme Value Distribution

2020 ◽  
Author(s):  
Arianna Miniussi ◽  
Marco Marani ◽  
Gabriele Villarini
Keyword(s):  
Tropical Cyclones ◽  
Heavy Rainfall ◽  
Daily Rainfall ◽  
Extreme Value Distribution ◽  
Value Theory ◽  
Extreme Value ◽  
Value Distribution ◽  
Single Component ◽  
Accurate Estimation ◽  
Rainfall Events

<p>Tropical Cyclones (TCs) represent a threat in several areas of the world, among which the Eastern/South-Eastern United States are one of the highly impacted regions. In addition to the frequently analyzed hazards related to the strong winds and storm surges associated with TCs, they are also responsible for heavy rainfall, which can affect areas located very far from the storm center. The accurate estimation of rainfall extremes is crucial in several TC-related impacts, such as engineering design of buildings and prevention/protection measures, flood mapping, risk estimation and mitigation, insurance and re-insurance purposes, policy-making support. Statistical approaches considering the physical drivers of hydrological phenomena, besides their conceptual relevance, can help reducing the estimation uncertainty of extremes. Under these premises, here we use the Metastatistical Extreme Value Distribution (MEVD), a recent approach that improves the estimation of high-return period values over the traditional Extreme Value Theory. We leverage the property of the MEVD to explicitly include in the statistical formulation different rainfall-generating phenomena and we examine the potential advantage of distinguishing TC-induced and non-TC rainfall events in the estimation of extremes. Hence, we apply the MEVD both in a single-component formulation (i.e., assuming that all rainfall events are generated by one single mechanism, so that they can be described by the same probability distribution) and a mixed-population formulation (i.e., separating non-TC and TC-induced rainfall events) to long time series of daily precipitation in six American metropolitan areas, historically known for being impacted by TCs. Moreover, due to the characteristic time scale of these mechanisms, which can significantly influence precipitation for several days, we focus also on aggregated values of rainfall on consecutive days. We find that the mixed approach is advantageous in some cases when looking at daily rainfall, especially when there is a rather uniform frequency of TC events over years. When considering cumulative rainfall on time windows of three days, we show that the reduction of the estimation error by the mixed MEVD is generally higher than in the case of daily rainfall and it is consistent for all the cases studied, except for Houston. A possible reason for the mixed MEVD not to outperform the single-component MEVD in this area is the presence of tornadic supercell convective mechanisms, which also generate heavy rainfall though concentrated in short time intervals.</p>

Extreme values of phase-type and mixed random variables with parallel-processing examples

1999 ◽  
Vol 36 (1) ◽  
pp. 194-210 ◽  
Author(s):  
Sungyeol Kang ◽  
Richard F. Serfozo
Keyword(s):  
Parallel Processing ◽  
Asymptotic Distribution ◽  
Extreme Values ◽  
Random Variables ◽  
Extreme Value Distribution ◽  
Rates Of Convergence ◽  
Value Theory ◽  
Extreme Value ◽  
Independent Random Variables ◽  
Phase Type

A basic issue in extreme value theory is the characterization of the asymptotic distribution of the maximum of a number of random variables as the number tends to infinity. We address this issue in several settings. For independent identically distributed random variables where the distribution is a mixture, we show that the convergence of their maxima is determined by one of the distributions in the mixture that has a dominant tail. We use this result to characterize the asymptotic distribution of maxima associated with mixtures of convolutions of Erlang distributions and of normal distributions. Normalizing constants and bounds on the rates of convergence are also established. The next result is that the distribution of the maxima of independent random variables with phase type distributions converges to the Gumbel extreme-value distribution. These results are applied to describe completion times for jobs consisting of the parallel-processing of tasks represented by Markovian PERT networks or task-graphs. In these contexts, which arise in manufacturing and computer systems, the job completion time is the maximum of the task times and the number of tasks is fairly large. We also consider maxima of dependent random variables for which distributions are selected by an ergodic random environment process that may depend on the variables. We show under certain conditions that their distributions may converge to one of the three classical extreme-value distributions. This applies to parallel-processing where the subtasks are selected by a Markov chain.

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