scholarly journals A class of probability distributions for application to non-negative annual maxima

2017 ◽  
Author(s):  
Earl Bardsley
Keyword(s):  
Weibull Distribution ◽  
Probability Distributions ◽  
Random Variables ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Value Distribution ◽  
Asymptotic Distributions ◽  
Generalized Extreme Value Distribution ◽  
Extreme Value Distributions ◽  
Specific Distribution

Abstract. Many environmental variables of interest as potential hazards take on only positive values, such a wind speed or river discharge. While recognising that primary interest is for largest extremes, it is desirable that distributions of maxima for design purposes should be subject to similar bounds as the physical variable concerned. A modified univariate extreme value argument defines a set of distributions, all bounded below at zero, with potential for application to annual maxima. Let f(x) be a probability distribution over the range, 0 ≤ x ≤ ω, where 0  0, c > 0 where ɛ = g(ω) and ɛ, ɑ, and c are respectively location, scale, and shape parameters. The distribution F(y) holds generally as an extreme value expression for sufficiently large N, irrespective of which of the three possible asymptotic extreme value distributions of sample maxima holds for X*. Therefore, the limit Weibull distribution for, say, Y* = X*−1 has no less validity as a single expression for obtaining exceedance probabilities than the generalized extreme value distribution applied directly to X*. If follows that a class of probability distributions for possible use with positive-valued annual maxima can be defined from the application of the inverse function g−1 to Weibull random variables for ɛ ≥ 0. All distributions so obtained are defined over the range 0 ≤ x ≤ ω, which actually excludes all of the asymptotic extreme value distributions of maxima except for the special case of the Type 2 extreme value distribution with location parameter at zero. It is to be expected, however, that the asymptotic distributions will sometimes hold to a high level of approximation within the 0, ω interval. No specific distribution is advocated for annual maxima application because concern here is only with drawing attention to the existence of the distribution class. The transformation approach is illustrated with respect the distribution of reciprocals of random variables generated from a 3-parameter Weibull distribution with ɛ ≥ 0.

Limit laws for maxima of a sequence of random variables defined on a Markov chain

1970 ◽  
Vol 2 (2) ◽  
pp. 323-343 ◽  
Author(s):  
Sidney I. Resnick ◽  
Marcel F. Neuts
Keyword(s):  
Markov Chain ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Value Distribution ◽  
Extreme Value Distributions ◽  
Limit Laws ◽  
Limiting Behavior ◽  
Normalizing Constants

Consider the bivariate sequence of r.v.'s {(Jn, Xn), n ≧ 0} with X0 = - ∞ a.s. The marginal sequence {Jn} is an irreducible, aperiodic, m-state M.C., m < ∞, and the r.v.'s Xn are conditionally independent given {Jn}. Furthermore P{Jn = j, Xn ≦ x | Jn − 1 = i} = pijHi(x) = Qij(x), where H1(·), · · ·, Hm(·) are c.d.f.'s. Setting Mn = max {X1, · · ·, Xn}, we obtain P{Jn = j, Mn ≦ x | J0 = i} = [Qn(x)]i, j, where Q(x) = {Qij(x)}. The limiting behavior of this probability and the possible limit laws for Mn are characterized.Theorem. Let ρ(x) be the Perron-Frobenius eigenvalue of Q(x) for real x; then:(a)ρ(x) is a c.d.f.;(b) if for a suitable normalization {Qijn(aijnx + bijn)} converges completely to a matrix {Uij(x)} whose entries are non-degenerate distributions then Uij(x) = πjρU(x), where πj = limn → ∞pijn and ρU(x) is an extreme value distribution;(c) the normalizing constants need not depend on i, j;(d) ρn(anx + bn) converges completely to ρU(x);(e) the maximum Mn has a non-trivial limit law ρU(x) iff Qn(x) has a non-trivial limit matrix U(x) = {Uij(x)} = {πjρU(x)} or equivalently iff ρ(x) or the c.d.f. πi = 1mHiπi(x) is in the domain of attraction of one of the extreme value distributions. Hence the only possible limit laws for {Mn} are the extreme value distributions which generalize the results of Gnedenko for the i.i.d. case.

Limit laws for maxima of a sequence of random variables defined on a Markov chain

1970 ◽  
Vol 2 (02) ◽  
pp. 323-343 ◽  
Author(s):  
Sidney I. Resnick ◽  
Marcel F. Neuts
Keyword(s):  
Markov Chain ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Value Distribution ◽  
Extreme Value Distributions ◽  
Limit Laws ◽  
Limiting Behavior ◽  
Normalizing Constants

Consider the bivariate sequence of r.v.'s {(J n , X n ), n ≧ 0} with X 0 = - ∞ a.s. The marginal sequence {J n } is an irreducible, aperiodic, m-state M.C., m &lt; ∞, and the r.v.'s X n are conditionally independent given {J n }. Furthermore P{J n = j, X n ≦ x | J n − 1 = i} = p ij H i (x) = Q ij (x), where H 1(·), · · ·, H m (·) are c.d.f.'s. Setting M n = max {X 1, · · ·, X n }, we obtain P{J n = j, M n ≦ x | J 0 = i} = [Q n (x)] i, j , where Q(x) = {Q ij (x)}. The limiting behavior of this probability and the possible limit laws for M n are characterized. Theorem. Let ρ(x) be the Perron-Frobenius eigenvalue of Q(x) for real x; then: (a)ρ(x) is a c.d.f.; (b) if for a suitable normalization {Q ij n (a ijn x + b ijn )} converges completely to a matrix {U ij (x)} whose entries are non-degenerate distributions then U ij (x) = π j ρ U (x), where π j = lim n → ∞ p ij n and ρ U (x) is an extreme value distribution; (c) the normalizing constants need not depend on i, j; (d) ρ n (a n x + b n ) converges completely to ρ U (x); (e) the maximum M n has a non-trivial limit law ρ U (x) iff Q n (x) has a non-trivial limit matrix U(x) = {U ij (x)} = {π j ρ U (x)} or equivalently iff ρ(x) or the c.d.f. π i = 1 m H i π i(x) is in the domain of attraction of one of the extreme value distributions. Hence the only possible limit laws for {M n } are the extreme value distributions which generalize the results of Gnedenko for the i.i.d. case.

scholarly journals Technical note: The Weibull distribution as an extreme value alternative for annual maxima

2018 ◽  
Author(s):  
Earl Bardsley
Keyword(s):  
Weibull Distribution ◽  
Extreme Value Distribution ◽  
Value Theory ◽  
Extreme Value ◽  
Technical Note ◽  
Value Distribution ◽  
Generalized Extreme Value Distribution ◽  
Sample Sizes ◽  
Large Sample ◽  
Incorrect Conclusion

Abstract. The generalized extreme value distribution (GEV) for largest extremes is widely applied to single-site annual maxima sequences for estimating exceedance probabilities for application to design magnitudes under conditions of stationarity. However, the GEV is not the only mode for application of classical extreme value theory to recorded maxima. An alternative approach is to apply specific transformations to the maxima, enabling different but equivalent exceedance statements. For example, the probability that annual flood maxima will exceed some magnitude e is the same as the probability that reciprocals of the maxima will be less than 1/ε. The transformed maxima considered here represent sample minima, where the sample is the number of transformed independent individual events per year. For sufficiently large sample sizes, this leads to just one of the extreme value distributions for design purposes – the Weibull distribution for minima. This extreme value distribution arises because it is the limit stable expression for describing distributions of large-sample minima when a lower bound is present. There is no way of telling whether a good Weibull fit to transformed annual maxima indicates that sample sizes are sufficiently large for the Weibull extreme value approximation to apply. It could happen that a good fit is simply a fortuitous empirical matching to data from transformation selection. However, a similar issue also applies to the GEV which is itself a flexible distribution capable of empirical matching to data. It is not possible to make a case for the Weibull distribution by application to a range of annual maxima because any number of different transformations might be applied to achieve good Weibull fits. Instead, two simple synthetic examples are used to illustrate how a good fit to annual maxima by the GEV could lead to an incorrect conclusion, in contrast to the Weibull approximation applied to the same examples.

Extreme Storm Surge estimates and projection through the Metastatistical Extreme Value Distribution

2021 ◽  
Author(s):  
Maria Francesca Caruso ◽  
Marco Marani
Keyword(s):  
Sea Level ◽  
Extreme Value Distribution ◽  
Storm Surges ◽  
Extreme Value ◽  
Value Distribution ◽  
Generalized Extreme Value Distribution ◽  
Sample Sizes ◽  
Mean Sea Level ◽  
Gev Distribution ◽  
Calibration Sample

&lt;p&gt;Storm surges caused by extreme meteorological conditions are a major natural risk in coastal areas, especially in the context of global climate change. The increase of future sea-levels caused by continuing global warming, may endanger human lives and infrastructure through inundation, erosion and salinization.&lt;br&gt;Hence, the reliable estimation of the occurrence probability of these extreme events is crucial to quantify risk and to design adequate coastal defense structures. The probability of event occurrence is typically estimated by modelling observed sea-level records using one of a few statistical approaches.&lt;br&gt;The traditional Extreme Value Theory is based on the use of the Generalized Extreme Value distribution (GEV), &amp;#160;fitted either by considering block (typically yearly) maxima, or values exceeding a high threshold (POT). This approach does not make full use of all observational information, and thereby does not minimize estimation uncertainty.&lt;br&gt;The recently proposed Metastatistical Extreme Value Distribution (MEVD), instead, makes use of most of the available observations and has been shown to outperform the classical GEV distribution in several applications, including hourly and daily rainfall, flood peak discharge and extreme hurricane intensity.&lt;br&gt;Here, we comparatively apply the MEVD and the GEV distribution to long time series of sea-level observations distributed along European coastlines (Venice (IT), Hornbaek (DK), Marseille (FR), Newlyn (UK)). A cross-validation approach, dividing available data in separate calibration and test sub-samples, is used to compare their performances in high-quantile estimation.&lt;br&gt;The MEVD approach is based on the definition of an &amp;#8220;ordinary values&amp;#8221; distribution (here a Generalized Pareto distribution), whose parameters are estimated using the Probability Weighted Moments method on non-overlapping sub-samples of fixed size (5 years). To address the problems posed by observational samples of different sizes, we explore the effect on uncertainty of different calibration sample sizes, from 5 to 30 years. In this application, we find that the GEVD-POT and MEVD approaches perform similarly, once the above parameter choices are optimized. In particular, when considering short samples (5 years) and events with a high return time, the estimation errors show no significant differences in their median value across methods and sites, all approaches producing a similar underestimation of the actual quantile. When larger calibration sample sizes are considered (10-30 yrs), the median error of MEVD estimates tends to be closer to zero than those obtained from the traditional methods.&lt;br&gt;Future projections of sea-level rise until 2100 are also analyzed, with reference to intermediate and extreme representative concentration pathways (RCP 4.5 and RCP 8.5). The probability of future storm surges along European coastlines are then estimated assuming a changing mean sea-level and an unchanged storm regime. The projections indicate future changes in mean sea-level lead to increases in the height of storm surges for a fixed return period that are spatially heterogeneous across the coastal locations explored.&lt;/p&gt;

Sign in / Sign up

Export Citation Format

Share Document