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Statistics of extremes

01 Jan 2011-Iss: 19, pp 1484-1487
TL;DR: This article reviews multivariate theory, distinguishing asymptotic independence and dependence models, followed by a description of models for spatial and spatiotemporal extreme events, and discusses inference and describe two applications.
Abstract: Statistics of extremes concerns inference for rare events. Often the events have never yet been observed, and their probabilities must therefore be estimated by extrapolation of tail models fitted to available data. Because data concerning the event of interest may be very limited, efficient methods of inference play an important role. This article reviews this domain, emphasizing current research topics. We first sketch the classical theory of extremes for maxima and threshold exceedances of stationary series. We then review multivariate theory, distinguishing asymptotic independence and dependence models, followed by a description of models for spatial and spatiotemporal extreme events. Finally, we discuss inference and describe two applications. Animations illustrate some of the main ideas.
Citations
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01 Apr 2013
TL;DR: In this paper, the authors investigated the presence of trends in annual maximum daily precipitation time series obtained from a global dataset of 8326 high-quality land-based observing stations with more than 30 years of record over the period from 1900 to 2009.
Abstract: This study investigates the presence of trends in annual maximum daily precipitation time series obtained from a global dataset of 8326 high-quality land-based observing stations with more than 30 years of record over the period from 1900 to 2009. Two complementary statistical techniques were adopted to evaluate the possible nonstationary behavior of these precipitation data. The first was a Mann‐Kendall nonparametric trend test, and it was used to evaluate the existence of monotonic trends. The second was a nonstationary generalized extreme value analysis, and it was used to determine the strength of association between the precipitation extremes and globally averaged near-surface temperature. The outcomes are that statistically significant increasing trends can be detected at the global scale, with close to two-thirds of stations showing increases. Furthermore, there is a statistically significant association with globally averaged near-surface temperature,withthemedianintensityofextremeprecipitationchanginginproportionwithchangesinglobal mean temperature at a rate of between 5.9% and 7.7%K 21 , depending on the method of analysis. This ratio was robust irrespective of record length or time period considered and was not strongly biased by the uneven global coverage of precipitation data. Finally, there is a distinct meridional variation, with the greatest sensitivity occurring in the tropics and higher latitudes and the minima around 138S and 118N. The greatest uncertainty was near the equator because of the limited number of sufficiently long precipitation records, and there remains an urgent need to improve data collection in this region to better constrain future changes in tropical precipitation.

615 citations

Journal ArticleDOI
TL;DR: In this paper, the main types of statistical models based on latent variables, on copulas and on spatial max-stable processes are described and compared by application to a data set on rainfall in Switzerland.
Abstract: The areal modeling of the extremes of a natural process such as rainfall or temperature is important in environmental statistics; for example, understanding extreme areal rainfall is crucial in flood protection. This article reviews recent progress in the statistical modeling of spatial extremes, starting with sketches of the necessary elements of extreme value statistics and geostatistics. The main types of statistical models thus far proposed, based on latent variables, on copulas and on spatial max-stable processes, are described and then are compared by application to a data set on rainfall in Switzerland. Whereas latent variable modeling allows a better fit to marginal distributions, it fits the joint distributions of extremes poorly, so appropriately-chosen copula or max-stable models seem essential for successful spatial modeling of extremes.

572 citations


Cites background from "Statistics of extremes"

  • ...Davison and Gholamrezaee (2012) fit models based on (22) and (23) to extreme temperature data....

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  • ...Padoan, Ribatet and Sisson (2010), Blanchet and Davison (2011) and Davison and Gholamrezaee (2012) discuss its application in the context of extremal inference, and its use to fit spatial extremal models based on (21) and (22) has been implemented in the R libraries SpatialExtremes and CompRandFld....

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Journal ArticleDOI
TL;DR: In this paper, the authors consider stochastic processes under resetting, which have attracted a lot of attention in recent years, and discuss multiparticle systems as well as extended systems, such as fluctuating interfaces.
Abstract: In this Topical Review we consider stochastic processes under resetting, which have attracted a lot of attention in recent years. We begin with the simple example of a diffusive particle whose position is reset randomly in time with a constant rate r, which corresponds to Poissonian resetting, to some fixed point (e.g. its initial position). This simple system already exhibits the main features of interest induced by resetting: (i) the system reaches a nontrivial nonequilibrium stationary state (ii) the mean time for the particle to reach a target is finite and has a minimum, optimal, value as a function of the resetting rate r. We then generalise to an arbitrary stochastic process (e.g. Levy flights or fractional Brownian motion) and non-Poissonian resetting (e.g. power-law waiting time distribution for intervals between resetting events). We go on to discuss multiparticle systems as well as extended systems, such as fluctuating interfaces, under resetting. We also consider resetting with memory which implies resetting the process to some randomly selected previous time. Finally we give an overview of recent developments and applications in the field. PACS numbers: 05.40.-a, 05.70.Fh, 02.50.Ey, 64.60.-i arXiv:1910.07993v2 [cond-mat.stat-mech]

361 citations

Journal ArticleDOI
TL;DR: In this article, the authors analyzed the annual maximum daily rainfall of 15,137 records from all over the world, with lengths varying from 40 to 163 years, and analyzed the fitting results focusing on the behavior of the shape parameter.
Abstract: [1] Theoretically, if the distribution of daily rainfall is known or justifiably assumed, then one could argue, based on extreme value theory, that the distribution of the annual maxima of daily rainfall would resemble one of the three limiting types: (a) type I, known as Gumbel; (b) type II, known as Frechet; and (c) type III, known as reversed Weibull. Yet, the parent distribution usually is not known and often only records of annual maxima are available. Thus, the question that naturally arises is which one of the three types better describes the annual maxima of daily rainfall. The question is of great importance as the naive adoption of a particular type may lead to serious underestimation or overestimation of the return period assigned to specific rainfall amounts. To answer this question, we analyze the annual maximum daily rainfall of 15,137 records from all over the world, with lengths varying from 40 to 163 years. We fit the generalized extreme value (GEV) distribution, which comprises the three limiting types as special cases for specific values of its shape parameter, and analyze the fitting results focusing on the behavior of the shape parameter. The analysis reveals that (a) the record length strongly affects the estimate of the GEV shape parameter and long records are needed for reliable estimates; (b) when the effect of the record length is corrected, the shape parameter varies in a narrow range; (c) the geographical location of the globe may affect the value of the shape parameter; and (d) the winner of this battle is the Frechet law.

291 citations


Cites background from "Statistics of extremes"

  • ...equivalently, that the underlying distribution is the Gumbel. It is highly probable for a null hypothesis with small values of , e.g., H0: 1⁄4 0.01 or H0: 1⁄4 0.01 not to be rejected. Hence, we deem that it is not possible to conclude with certainty applying statistical tests whether the underlying distribution is Gumbel or GEV with close to zero. [29] Nevertheless, apart from the aforementioned tests, graphical tools exist that are especially useful when dealing with a large number of records, which can help to make inference about the underlying distribution. A graphical tool that has gained popularity over the last decade, introduced by Hosking [1990], is provided by the L-moments ratio diagrams....

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  • ...…has been an active research field and a matter of debate for more than half a century dating back to the works of E. J. Gumbel in 1940s; however, the field of extreme value theory seems to have originated more than three centuries ago in the works of Nicolaus Bernoulli [see e.g., Gumbel, 1958]....

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Book
04 Nov 2010
TL;DR: In this article, the authors introduce persistence models and Bootstrap Confidence Intervals for univariate and bivariate time series analysis, and present a future direction for future directions. But, they do not discuss the use of spectral analysis.
Abstract: Part I: Fundamental Concepts.- 1 Introduction.- 2 Persistence Models.- 3 Bootstrap Confidence Intervals.- Part II: Univariate Time Series.- 4 Regression I.- 5 Spectral Analysis.- 6. Extreme Value Time Series.- Part III: Bivariate Time Series.- 7 Correlation.- 8 Regression II.- Part IV: Outlook.- 9 Future Directions.

261 citations

References
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Journal ArticleDOI
01 Apr 1928
TL;DR: In this article, the problem of finding the appropriate limiting distribution in any case may be found from the manner in which the probability of exceeding any value x tends to zero as x is increased.
Abstract: The limiting distribution, when n is large, of the greatest or least of a sample of n, must satisfy a functional equation which limits its form to one of two main types. Of these one has, apart from size and position, a single parameter h, while the other is the limit to which it tends when h tends to zero.The appropriate limiting distribution in any case may be found from the manner in which the probability of exceeding any value x tends to zero as x is increased. For the normal distribution the limiting distribution has h = 0.From the normal distribution the limiting distribution is approached with extreme slowness; the final series of forms passed through as the ultimate form is approached may be represented by the series of limiting distributions in which h tends to zero in a definite manner as n increases to infinity.Numerical values are given for the comparison of the actual with the penultimate distributions for samples of 60 to 1000, and of the penultimate with the ultimate distributions for larger samples.

3,079 citations

Journal ArticleDOI
TL;DR: In this paper, a simple general approach to inference about the tail behavior of a distribution is proposed, which is not required to assume any global form for the distribution function, but merely the form of behavior in the tail where it is desired to draw inference.
Abstract: A simple general approach to inference about the tail behavior of a distribution is proposed. It is not required to assume any global form for the distribution function, but merely the form of behavior in the tail where it is desired to draw inference. Results are particularly simple for distributions of the Zipf type, i.e., where $G(y) = 1 - Cy^{-\alpha}$ for large $y$. The methods of inference are based upon an evaluation of the conditional likelihood for the parameters describing the tail behavior, given the values of the extreme order statistics, and can be implemented from both Bayesian and frequentist viewpoints.

3,060 citations


"Statistics of extremes" refers background in this paper

  • ...This approach is equivalent to attributing an approximate exponential distribution with mean ξ to large log Yj , and it yields the celebrated Hill (1975) estimator ξ̂H,k = k−1 k∑ j=1 log[Y (n+1− j )/Y (n−k)], (29) where Y (1) ≤ · · · ≤ Y (n) denote the ordered Yj ....

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Journal ArticleDOI

2,037 citations


"Statistics of extremes" refers background in this paper

  • ...The identification of limiting distributions for maxima is attributable to Fisher & Tippett (1928), whose work was further developed by von Mises (1936), Gnedenko (1943), and many others....

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  • ...This question was addressed by von Mises (1936) and particularly by Gnedenko (1943), who provided necessary and sufficient conditions for F ∈ MDAξ in terms of the upper tail properties of F (de Haan & Ferreira 2006, chapter 1)....

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Journal ArticleDOI
TL;DR: In this article, the authors discuss the analysis of the extremes of data by modelling the sizes and occurrence of exceedances over high thresholds, and the natural distribution for such exceedances, the generalized Pareto distribution, is described and its properties elucidated.
Abstract: We discuss the analysis of the extremes of data by modelling the sizes and occurrence of exceedances over high thresholds. The natural distribution for such exceedances, the generalized Pareto distribution, is described and its properties elucidated. Estimation and model-checking procedures for univariate and regression data are developed, and the influence of and information contained in the most extreme observations in a sample are studied. Models for seasonality and serial dependence in the point process of exceedances are described. Sets of data on river flows and wave heights are discussed, and an application to the siting of nuclear installations is described

1,503 citations

Journal ArticleDOI
TL;DR: In this paper, the authors use the method of probability-weighted moments to derive estimators of the parameters and quantiles of the generalized extreme-value distribution, and investigate the properties of these estimators in large samples via asymptotic theory, and in small and moderate samples, via computer simulation.
Abstract: We use the method of probability-weighted moments to derive estimators of the parameters and quantiles of the generalized extreme-value distribution. We investigate the properties of these estimators in large samples, via asymptotic theory, and in small and moderate samples, via computer simulation. Probability-weighted moment estimators have low variance and no severe bias, and they compare favorably with estimators obtained by the methods of maximum likelihood or sextiles. The method of probability-weighted moments also yields a convenient and powerful test of whether an extreme-value distribution is of Fisher-Tippett Type I, II, or III.

1,275 citations