Gumbel Extreme Value Distribution Research Articles

Testing for the change of the mean-reverting parameter of an autoregressive model with stationary Gaussian noise

The likelihood ratio test for a change in the mean-reverting parameter of a first order autoregressive model with stationary Gaussian noise is considered. The test statistic converges in distribution to the Gumbel extreme value distribution under the null hypothesis of no change-point for a large class of covariance structures including long-memory processes as the fractional Gaussian noise.

Universal gap scaling in percolation

Universality is a principle that fundamentally underlies many critical phenomena, ranging from epidemic spreading to the emergence or breakdown of global connectivity in networks. Percolation, the transition to global connectedness on gradual addition of links, may exhibit substantial gaps in the size of the largest connected network component. We uncover that the largest gap statistics is governed by extreme-value theory. This allows us to unify continuous and discontinuous percolation by virtue of universal critical scaling functions, obtained from normal and extreme-value statistics. Specifically, we show that the universal scaling function of the size of the largest gap is given by the extreme-value Gumbel distribution. This links extreme-value statistics to universality and criticality in percolation. Percolation transitions underpin a generic class of phenomena associated with the degree of connectedness in networks. A detailed numerical study now uncovers a universal scaling in the size of the largest cluster identified in such percolation models.

The limit distribution of the maximum probability nearest-neighbour ball

AbstractLet X1, …, Xn be independent random points drawn from an absolutely continuous probability measure with density f in ℝd. Under mild conditions on f, wederive a Poisson limit theorem for the number of large probability nearest-neighbour balls. Denoting by Pn the maximum probability measure of nearest-neighbour balls, this limit theorem implies a Gumbel extreme value distribution for nPn − ln n as n → ∞. Moreover, we derive a tight upper bound on the upper tail of the distribution of nPn − ln n, which does not depend on f.

Predicting Maximal Gaps in Sets of Primes

Let q > r ≥ 1 be coprime integers. Let P c = P c ( q , r , H ) be an increasing sequence of primes p satisfying two conditions: (i) p ≡ r (mod q) and (ii) p starts a prime k-tuple with a given pattern H. Let π c ( x ) be the number of primes in P c not exceeding x. We heuristically derive formulas predicting the growth trend of the maximal gap G c ( x ) = max p ′ ≤ x ( p ′ − p ) between successive primes p , p ′ ∈ P c. Extensive computations for primes up to 10 14 show that a simple trend formula G c ( x ) ∼ x π c ( x ) · ( log π c ( x ) + O k ( 1 ) ) works well for maximal gaps between initial primes of k-tuples with k ≥ 2 (e.g., twin primes, prime triplets, etc.) in residue class r (mod q). For k = 1, however, a more sophisticated formula G c ( x ) ∼ x π c ( x ) · log π c 2 ( x ) x + O ( log q ) gives a better prediction of maximal gap sizes. The latter includes the important special case of maximal gaps in the sequence of all primes (k = 1 , q = 2 , r = 1). The distribution of appropriately rescaled maximal gaps G c ( x ) is close to the Gumbel extreme value distribution. Computations suggest that almost all maximal gaps satisfy a generalized strong form of Cramér’s conjecture. We also conjecture that the number of maximal gaps between primes in P c below x is O k ( log x ).

Assessment of Flood Extremes Using Downscaled CMIP5 High-Resolution Ensemble Projections of Near-Term Climate for the Upper Thu Bon Catchment in Vietnam

Exploring potential floods is both essential and critical to making informed decisions for adaptation options at a river basin scale. The present study investigates changes in flood extremes in the future using downscaled CMIP5 (Coupled Model Intercomparison Project—Phase 5) high-resolution ensemble projections of near-term climate for the Upper Thu Bon catchment in Vietnam. Model bias correction techniques are utilized to improve the daily rainfall simulated by the multi-model climate experiments. The corrected rainfall is then used to drive a calibrated supper-tank model for runoff simulations. The flood extremes are analyzed based on the Gumbel extreme value distribution and simulation of design hydrograph methods. Results show that the former method indicates almost no changes in the flood extremes in the future compared to the baseline climate. However, the later method explores increases (approximately 20%) in the peaks of very extreme events in the future climate, especially, the flood peak of a 50-year return period tends to exceed the flood peak of a 100-year return period of the baseline climate. Meanwhile, the peaks of shorter return period floods (e.g., 10-year) are projected with a very slight change. Model physical parameterization schemes and spatial resolution seem to cause larger uncertainties; while different model runs show less sensitivity to the future projections.

Statistical analysis of high cycle fatigue life and inclusion size distribution in shot peened 300M steel

In order to fully exploit the benefits of shot peening process in terms of fatigue performance, the proper choice of peening conditions is indispensable, which is largely material specific. In this study, the influence of 3 different peening conditions on the axial fatigue life of high strength 300M steel (modified AISI 4340) was investigated in the high cycle fatigue regime. A statistical estimate of 90% survival life at 95% confidence interval was made to rigorously compare the fatigue performance of the material with different surface conditions. The peening induced roughness, compressive residual stresses and their relaxation during cyclic loading were quantified. It was however revealed, that shot peening had no significant effect on this material when tested at 55% of yield strength (σy,0.2). The fatigue life was dominated by the larger non-metallic inclusions within the bulk material. An empirical relationship between inclusion size and fatigue life was proposed based on this observation. Furthermore, the inclusion size distribution was well described using Gumbel extreme value distribution allowing the largest inclusion to be predicted in a given volume of steel. The prediction results were in good agreement with the experimentally observed value of the largest inclusion size observed at a fatigue crack initiation site.

Rainfall–Runoff Analysis for Sustainable Stormwater Drainage for the City of Madinah, Saudi Arabia

Design of urban drainage infrastructure depends on the rainfall pattern and runoff volumes. Under the current impacts of urbanization and industrialization in the Kingdome of Saudi Arabia (KSA), there is a need to review the guidelines for design of drainage infrastructure. In this study, a methodology is developed to investigate the impact of variations in rainfall on the capacity of drainage infrastructure for the city of Madinah (Taibah), KSA. Rainfall data collected from Madinah Municipality was analyzed to determine the normal annual rainfall (average of the annual rainfall for 30 years). Data consistency was checked using double mass curve technique, and the design rainfall (DR) was estimated by Gumbel Extreme Value Distribution. The Mann-Kendall test was run at 5% significance level on rainfall time series over the period 1985–2015; however, no significant trend was observed. Urban drainage schemes are designed for peak flows based on DR. The true values of DR are difficult to estimate due to uncertainties associated with variations in estimation procedures and data limitations. An attempt is made to elaborate the impact of data errors (both systematic and random) on DR to facilitate engineers in selecting safety factor for design of urban drainage infrastructure. The stormwater trunk-sewer was designed to safely convey the runoff for an area around the Masjid-e-Al-Nabawi. Rainfall–runoff modeling was performed by Rational Method to find the peak flow and Nash GIUH. It was found that there is about 20% change in diameter of trunk sewer for change in return period from 2 to 10 years.

Higher order expansion for moments of extreme for generalized Maxwell distribution

ABSTRACTIn this paper, the higher order asymptotic expansion of the moments of extreme from generalized Maxwell distribution is gained, by which one establishes the rate of convergence of the moment of the normalized partial maximum to the moment of the associated Gumbel extreme value distribution.

Extreme value statistics for pitting corrosion of old underground cast iron pipes

Many major city water supply distribution networks consist of buried cast iron pipes. In many cases the pipes are internally cement-lined and the predominant corrosion is by external pitting. This may cause leakage and eventual structural failure. It is conventional to use the Gumbel extreme value distribution to represent the statistics of maximum pits depth and to use it to estimate the probability of pipe wall perforation. Herein data obtained for maximum pit depths for large-sized (1–2m long) samples of 10 pipes exhumed from different, apparently randomly selected, locations after 34–129 years of service are examined for consistency with the Gumbel probability distribution. This was the case for the deepest pits, but the data for less deep pits show a consistent pattern of departure from the Gumbel distribution. Some extreme pit depth data, inconsistent with the rest are interpreted as possibly caused by material imperfections.

A comparative review of generalizations of the Gumbel extreme value distribution with an application to wind speed data

ABSTRACTThe generalized extreme value distribution and its particular case, the Gumbel extreme value distribution, are widely applied for extreme value analysis. The Gumbel distribution has certain drawbacks because it is a non-heavy-tailed distribution and is characterized by constant skewness and kurtosis. The generalized extreme value distribution is frequently used in this context because it encompasses the three possible limiting distributions for a normalized maximum of infinite samples of independent and identically distributed observations. However, the generalized extreme value distribution might not be a suitable model when each observed maximum does not come from a large number of observations. Hence, other forms of generalizations of the Gumbel distribution might be preferable. Our goal is to collect in the present literature the distributions that contain the Gumbel distribution embedded in them and to identify those that have flexible skewness and kurtosis, are heavy-tailed and could be competitive with the generalized extreme value distribution. The generalizations of the Gumbel distribution are described and compared using an application to a wind speed data set and Monte Carlo simulations. We show that some distributions suffer from overparameterization and coincide with other generalized Gumbel distributions with a smaller number of parameters, that is, are non-identifiable. Our study suggests that the generalized extreme value distribution and a mixture of two extreme value distributions should be considered in practical applications.

Asymptotic properties for distributions and densities of extremes from generalized gamma distribution

For a generalized gamma random sequence, some fundamental properties and convergence rates of the distribution of its partial maximum to the Gumbel extreme value distribution are derived. The asymptotic expansions for the distribution and density of maximum from generalized gamma distribution are given under an optimal choice of normalizing constants.

Records for the number of distinct sites visited by a random walk on the fully connected lattice

We consider a random walk on the fully connected lattice with N sites and study the time evolution of the number of distinct sites s visited by the walker on a subset with n sites. A record value v is obtained for s at a record time t when the walker visits a site of the subset for the first time. The record time t is a partial covering time when and a total covering time when v = n. The probability distributions for the number of records s, the record value v and the record (covering) time t, involving r-Stirling numbers, are obtained using generating function techniques. The mean values, variances and skewnesses are deduced from the generating functions. In the scaling limit the probability distributions for s and v lead to the same Gaussian density. The fluctuations of the record time t are also Gaussian at partial covering, when They are distributed according to the type-I Gumbel extreme-value distribution at total covering, when v = n. A discrete sequence of generalized Gumbel distributions, indexed by is obtained at almost total covering, when These generalized Gumbel distributions are crossing over to the Gaussian distribution when n − v increases.

Higher-order expansions for distributions of extremes from general error distribution

In this short note, with optimal normalizing constants, the higher-order expansion for a distribution of normalized partial maximum from the general error distribution is derived, by which one deduces the associate convergence rate of the distribution of the extreme to the Gumbel extreme value distribution. MSC:62E20, 60E05, 60F15, 60G15.

The Distribution of Maximal Prime Gaps in Cramer's Probabilistic Model of Primes

In the framework of Cramer's probabilistic model of primes, we explore the exact and asymptotic distributions of maximal prime gaps. We show that the Gumbel extreme value distribution exp(-exp(-x)) is the limit law for maximal gaps between Cramer's random primes. The result can be derived from a general theorem about intervals between discrete random events occurring with slowly varying probability monotonically decreasing to zero. A straightforward generalization extends the Gumbel limit law to maximal gaps between prime constellations in Cramer's model.

A spectral theory approach for extreme value analysis in a tandem of fluid queues

We consider a model to evaluate performance of streaming media over an unreliable network. Our model consists of a tandem of two fluid queues. The first fluid queue is a Markov modulated fluid queue that models the network congestion, and the second queue represents the play-out buffer. For this model the distribution of the total amount of fluid in the congestion and play-out buffer corresponds to the distribution of the maximum attained level of the first buffer. We show that, under proper scaling and when we let time go to infinity, the distribution of the total amount of fluid converges to a Gumbel extreme value distribution. From this result, we derive a simple closed-form expression for the initial play-out buffer level that provides a probabilistic guarantee for undisturbed play-out.

Statistical Foundation of Empirical Isotherms

We show that most of the empirical or semi-empirical isotherms proposed to extend the Langmuir formula to sorption (adsorption, chimisorption and biosorption) on heterogeneous surfaces in the gaseous and liquid phase belong to the family and subfamily of the BurrXII cumulative distribution functions. As a consequence they obey relatively simple differential equations which describe birth and death phenomena resulting from mesoscopic and microscopic physicochemical processes. Using the probability theory, it is thus possible to give a physical meaning to their empirical coefficients, to calculate well defined quantities and to compare the results obtained from different isotherms. Another interesting consequence of this finding is that it is possible to relate the shape of the isotherm to the distribution of sorption energies which we have calculated for each isotherm. In particular, we show that the energy distribution corresponding to the Brouers-Sotolongo (BS) isotherm [1] is the Gumbel extreme value distribution. We propose a generalized GBS isotherm, calculate its relevant statistical properties and recover all the previous results by giving well defined values to its coefficients. Finally we show that the Langmuir, the Hill-Sips, the BS and GBS isotherms satisfy the maximum Bolzmann-Shannon entropy principle and therefore should be favoured.

Rates of convergence of extremes from skew-normal samples

For a skew-normal random sequence, convergence rates of the distribution of its partial maximum to the Gumbel extreme value distribution are derived. The asymptotic expansion of the distribution of the normalized maximum is given under an optimal choice of norming constants. We find that the optimal convergence rate of the normalized maximum to the Gumbel extreme value distribution is proportional to 1/logn.

Tail Properties and Asymptotic Expansions for the Maximum of the Logarithmic Skew-Normal Distribution

We discuss tail behaviors, subexponentiality, and the extreme value distribution of logarithmic skew-normal random variables. With optimal normalized constants, the asymptotic expansion of the distribution of the normalized maximum of logarithmic skew-normal random variables is derived. We show that the convergence rate of the distribution of the normalized maximum to the Gumbel extreme value distribution is proportional to 1/(log n)1/2.

Tail Properties and Asymptotic Expansions for the Maximum of the Logarithmic Skew-Normal Distribution

We discuss tail behaviors, subexponentiality, and the extreme value distribution of logarithmic skew-normal random variables. With optimal normalized constants, the asymptotic expansion of the distribution of the normalized maximum of logarithmic skew-normal random variables is derived. We show that the convergence rate of the distribution of the normalized maximum to the Gumbel extreme value distribution is proportional to 1/(log n)1/2.

Gumbel fluctuations for cover times in the discrete torus

This work proves that the fluctuations of the cover time of simple random walk in the discrete torus of dimension at least three with large side-length are governed by the Gumbel extreme value distribution. This result was conjectured for example in Aldous and Fill (Reversible Markov chains and random walks on graphs, in preparation). We also derive some corollaries which qualitatively describe “how” covering happens. In addition, we develop a new and stronger coupling of the model of random interlacements, introduced by Sznitman (Ann Math (2) 171(3):2039–2087, 2010), and random walk in the torus. This coupling is used to prove the cover time result and is also of independent interest.