Laws of Small Numbers: Extremes and Rare Events

Falk, Michael; Hüsler, Jürg; Reiss, Rolf–Dieter

doi:10.1007/978-3-0348-7791-6

Cited by 232 publications

(195 citation statements)

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“…random variables is by now rather standard (Castillo 1988;Coles 2001;Embrechts et al 1997;Falk et al 1994;Galambos 1978;Gumbel 1958;Jenkinson 1955;Lindgren et al 1983;Reiss and Thomas 2001;Tiago de Oliveira 1984). The foundation is Gnedenko's theorem (Gnedenko 1943;Fisher and Tippett 1928).…”

Section: Gev Modelling For Non-stationary Time Series 3a Stationary mentioning

confidence: 99%

Extreme Value Statistics of the Total Energy in an Intermediate-Complexity Model of the Midlatitude Atmospheric Jet. Part I: Stationary Case

Felici

Lucarini

Speranza

et al. 2007

Journal of the Atmospheric Sciences

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A baroclinic model for the atmospheric jet at middle-latitudes is used as a stochastic generator of non-stationary time series of the total energy of the system. A linear time trend is imposed on the parameter T E , descriptive of the forced equator-to-pole temperature gradient and responsible for setting the average baroclinicity in the model. The focus lies on establishing a theoretically sound framework for the detection and assessment of trend at extreme values of the generated time series. This problem is dealt with by fitting time-dependent Generalized Extreme Value (GEV) models to sequences of yearly maxima of the total energy. A family of GEV models is used in which the location µ and scale parameters σ depend quadratically and linearly on time, respectively, while the shape parameter ξ is kept constant. From this family, a model is selected by using diagnostic graphical tools, such as probability and quantile plots, and by means of the likelihood ratio test. The inferred location and scale parameters are found to depend in a rather smooth way on time and, therefore, on T E . In particular, power-law dependences of µ and σ on T E are obtained, in analogy with the results of a previous work where the same baroclinic model was run with fixed values of T E spanning the same range as in this case. It is emphasized under which conditions the adopted approach is valid.

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Section: Gev Modelling For Non-stationary Time Series 3a Stationary mentioning

confidence: 99%

Extreme Value Statistics of the Total Energy in an Intermediate-Complexity Model of the Midlatitude Atmospheric Jet. Part I: Stationary Case

Felici

Lucarini

Speranza

et al. 2007

Journal of the Atmospheric Sciences

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“…To the best of the authors' knowledge, the present paper is the first to establish such a limit theorem for jump processes. Compound Poisson limits for exceedances of a high level by a stationary Markov chain were obtained in [29]; such limits for exceedances and also for upcrossings of sequences are summarised in [13]. A general compound Poisson limit theorem for stationary, strongly mixing random measures was derived in [23].…”

Section: Introductionmentioning

confidence: 99%

On level crossings for a general class of piecewise-deterministic Markov processes

Borovkov

2008

Adv. Appl. Probab.

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We consider a piecewise-deterministic Markov process (X t ) governed by a jump intensity function, a rate function that determines the behaviour between jumps, and a stochastic kernel describing the conditional distribution of jump sizes. The paper deals with the point process N b + of upcrossings of some level b by (X t ). We prove a version of Rice's formula relating the stationary density of (X t ) to level crossing intensities and show that, for a wide class of processes (X t ), as b → ∞, the scaled point process (N b, where ν + (b) denotes the intensity of upcrossings of b, converges weakly to a geometrically compound Poisson process.

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“…This is the peaks over threshold approach [see Reiss and Thomas (2007) or Theorem 1.3.5 of Falk et al (2004) for more details]. In our case the exponential distribution turned out to be the appropriate choice of a GPD.…”

Section: The Algorithmmentioning

confidence: 73%

A new detection algorithm for image analysis of single, fluorescence‐labeled proteins in living cells

Michel

Steinmeyer

Falk

et al. 2007

Microscopy Res & Technique

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KEY WORDSpeaks over threshold; single-molecule microscopy; single-molecule tracking ABSTRACT A new algorithm is presented for the detection of single, fluorescence-labeled proteins in the analysis of images from living cells. It is especially suited for images with just a few (<1 per 10 lm 2 ) fluorescence peaks from individual proteins with high background and noise (signal to background ratios as low as 2 and signal to noise as low as 10). The analysis uses the peaks over threshold method from extreme value theory and requires minimal assumptions on the underlying distributions. The significant advantage of the method over others is the rare occurrence to detect false positives. Some examples of simulated and real data are given as comparisons. The algorithm is implemented in MATLAB. Microsc. Res. Tech. 70:763-770, 2007. V

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Laws of Small Numbers: Extremes and Rare Events

Cited by 232 publications

References 0 publications

Extreme Value Statistics of the Total Energy in an Intermediate-Complexity Model of the Midlatitude Atmospheric Jet. Part I: Stationary Case

Extreme Value Statistics of the Total Energy in an Intermediate-Complexity Model of the Midlatitude Atmospheric Jet. Part I: Stationary Case

On level crossings for a general class of piecewise-deterministic Markov processes

A new detection algorithm for image analysis of single, fluorescence‐labeled proteins in living cells

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