Extremes and Related Properties of Random Sequences and Processes

Leadbetter, M. R.; Lindgren, Georg; Rootzén, Holger

doi:10.1007/978-1-4612-5449-2

Cited by 2,722 publications

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“…Equally important, the analysis does not apply to extreme precipitation, such as precipitation above the 99.9th percentile (e.g., Groisman et al 2005), for which the theory of statistical extremes more likely provides the appropriate description (e.g., Leadbetter et al 1983;Meehl et al 2000;Wilson and Toumi 2005).…”

Section: Discussionmentioning

confidence: 99%

A Possible Constraint on Regional Precipitation Intensity Changes under Global Warming

Gutowski

Takle

Kozak

et al. 2007

Journal of Hydrometeorology

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Changes in daily precipitation versus intensity under a global warming scenario in two regional climate simulations of the United States show a well-recognized feature of more intense precipitation. More important, by resolving the precipitation intensity spectrum, the changes show a relatively simple pattern for nearly all regions and seasons examined whereby nearly all high-intensity daily precipitation contributes a larger fraction of the total precipitation, and nearly all low-intensity precipitation contributes a reduced fraction. The percentile separating relative decrease from relative increase occurs around the 70th percentile of cumulative precipitation, irrespective of the governing precipitation processes or which model produced the simulation. Changes in normalized distributions display these features much more consistently than distribution changes without normalization.Further analysis suggests that this consistent response in precipitation intensity may be a consequence of the intensity spectrum's adherence to a gamma distribution. Under the gamma distribution, when the total precipitation or number of precipitation days changes, there is a single transition between precipitation rates that contribute relatively more to the total and rates that contribute relatively less. The behavior is roughly the same as the results of the numerical models and is insensitive to characteristics of the baseline climate, such as average precipitation, frequency of rain days, and the shape parameter of the precipitation's gamma distribution. Changes in the normalized precipitation distribution give a more consistent constraint on how precipitation intensity may change when climate changes than do changes in the nonnormalized distribution. The analysis does not apply to extreme precipitation for which the theory of statistical extremes more likely provides the appropriate description. ABSTRACT Changes in daily precipitation versus intensity under a global warming scenario in two regional climate simulations of the United States show a well-recognized feature of more intense precipitation. More important, by resolving the precipitation intensity spectrum, the changes show a relatively simple pattern for nearly all regions and seasons examined whereby nearly all high-intensity daily precipitation contributes a larger fraction of the total precipitation, and nearly all low-intensity precipitation contributes a reduced fraction. The percentile separating relative decrease from relative increase occurs around the 70th percentile of cumulative precipitation, irrespective of the governing precipitation processes or which model produced the simulation. Changes in normalized distributions display these features much more consistently than distribution changes without normalization.Further analysis suggests that this consistent response in precipitation intensity may be a consequence of the intensity spectrum's adherence to a gamma distribution. Under the gamma distribution, when the total precipitation or number of p...

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Section: Discussionmentioning

confidence: 99%

A Possible Constraint on Regional Precipitation Intensity Changes under Global Warming

Gutowski

Takle

Kozak

et al. 2007

Journal of Hydrometeorology

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“…Following Leadbetter et al (1983) we refer to these conditions as D(u n ) and D (u n ), where u n is a suitable sequence of thresholds converging to sup ω∈Ω X(ω) = 1, as n goes to ∞, that will be defined below. D(u n ) imposes a certain type of distributional mixing property.…”

Section: Introductionmentioning

confidence: 99%

On the link between dependence and independence in extreme value theory for dynamical systems

Freitas

2008

Statistics & Probability Letters

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Abstract. It is well known that under some conditions on the dependence structure we can relate the asymptotic distribution of the partial maximum of a stationary stochastic process with the maximum of an associated independent sequence of random variables with the same distribution function of the dependent one. These conditions are known as D(u n ) and D (u n ). Although D(u n ) is of mixing type, when studying stochastic processes arising from a dynamical system with good mixing properties, verifying D(u n ) is not straightforward. We propose a reformulation of D(u n ) so that its validity may follow easily if we have a certain decay of correlations for the dynamical system in consideration.

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“…Consider a candidate pair fe i 1 ,t 1 ,e i 2 ,t 2 g. In lieu of (2) property (10) implies pairwise (5) and (6) the e i,t 's are serially and mutually asymptotically independent (Loynes, 1965;Ledford and Tawn, 1996, 2003.…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…Another tail exponent dependence measure is due to Leadbetter (1974Leadbetter ( , 1983, cf. Leadbetter et al (1983). The maximum max 1 r t r n jx t j of many weakly dependent sequences {x t } t = 1 n satisfies lim n-1 P 1 b n max 1 r t r n jx t jr z !…”

Section: Introductionmentioning

confidence: 99%

“…Notice if (2) holds for some r(h) then lim z-1 Pðx t 4 zjx tÀh 4 zÞ ¼ ð1 þrðhÞÞ Â lim z-1 Pðx t 4 zÞ ¼ 0, hence SV is asymptotically independent. Ledford and Tawn (1996, 2003 propose a bivariate tail model to give more structure to the asymptotic independence case. Define a unit Fré chet transformation x Ã t :¼ À1=lnF t ðx t Þ where F t ðxÞ :¼ Pðx t rxÞ, thus Pðx Ã t r zÞ ¼ expfÀ1=zg.…”

Section: Introductionmentioning

confidence: 99%

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Extremal memory of stochastic volatility with an application to tail shape inference

Hill

2011

Journal of Statistical Planning and Inference

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a b s t r a c tWe characterize joint tails and tail dependence for a class of stochastic volatility processes. We derive the exact joint tail shape of multivariate stochastic volatility with innovations that have a regularly varying distribution tail. This is used to give four new characterizations of tail dependence. In three cases tail dependence is a non-trivial function of linear volatility memory parametrically represented by tail scales, while tail power indices do not provide any relevant dependence information. Although tail dependence is associated with linear volatility memory, tail dependence itself is nonlinear. In the fourth case a linear function of tail events and exceedances is linearly independent. Tail dependence falls in a class that implies the celebrated Hill (1975) tail index estimator is asymptotically normal, while linear independence of nonlinear tail arrays ensures the asymptotic variance is the same as the iid case. We illustrate the latter finding by simulation.

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Extremes and Related Properties of Random Sequences and Processes

Cited by 2,722 publications

References 0 publications

A Possible Constraint on Regional Precipitation Intensity Changes under Global Warming

A Possible Constraint on Regional Precipitation Intensity Changes under Global Warming

On the link between dependence and independence in extreme value theory for dynamical systems

Extremal memory of stochastic volatility with an application to tail shape inference

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