Statistical Inference Using Extreme Order Statistics

By J. Pickands. 1975

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Citations in the corpus

A Wild Bootstrap for Degenerate Kernel Tests (2014, methodology): “To model the tail of this distribution, we have fitted the generalized Pareto distribution to the bootstrapped samples ([23] shows that for a large class of underlying distribution functions such an approximation is valid).”
The Extreme Value Machine (2015, background): “ume only limited forms. To ﬁnd the appropriate form, let us ﬁrst recall: 1There are other types of extreme value theorems, e.g., the second extreme value theorem, the Pickands-Balkema-de Haan Theorem [36], addresses probabilities conditioned on the process exceeding a sufﬁciently high threshold. The”
On the emergence of rainfall extremes from ordinary events (2016, methodology): “A second approach, termed peak over threshold (POT) method [Balkema and Haan, 1974; Pickands, 1975] (i) fixes a high intensity threshold, q; (ii) assumes a Poisson occurrence of events above the threshold; and (iii) models the excess values over q (assumed to be independent of the occurrence…”
A Tail-Index Analysis of Stochastic Gradient Noise in Deep Neural Networks (2019, background): “Some of the well-known estimators for this task are (Hill, 1975; Pickands, 1975; Dekkers et al., 1989; De Haan & Peng, 1998).”
Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach (2000, background): “Balkema and de Haan (1974) and Pickands (1975) showed for a large class of distributions F that it is possible to nd a positive measurable function (u) such that”
Nonparametric regression estimation of conditional tails: the random covariate case (2014, methodology): “Daouia et al. [17] studied the estimation of extreme quantiles under a conditional Paretotype model with random covariates, and used a fixed number of such quantile estimates in classical estimators for γ , like the Pickands [18] and the Hill [19] estimators.”
On maximum likelihood estimation of the extreme value index (2004, unknown): “Then it is well known [see Balkema and de Haan (1974) and Pickands (1975)] that up to scale and location transformations the generalized Pareto d.f. given by Hγ (x) = 1− (1+ γ x)−1/γ ,(2) x > 0 if γ ≥ 0 and 0 x −1/γ if γ 0 [for γ = 0 read(1 + γ x)−1/γ as exp(−x)], can provide a good…”
On testing extreme value conditions (2006, background): “Under this condition the common approach usually estimates the extreme value index γ and normalizing constants an and b n for a given sample of size n (e.g., Hill (1975), Pickands (1975), Hall (1982), Smith (1985) and Dekkers et al. (1989)).”
Estimating Local Intrinsic Dimensionality (2015, background): “The Fisher-Tippett-Gnedenko Theorem and the PickandsBalkema-de Haan Theorem have been shown to be equivalent to a third characterization of the tail behavior, in terms of regularly-varying (RV) functions.”
A two-step framework for over-threshold modelling of environmental extremes (2014, methodology): “Note also that, following the statistical theory (Pickands, 1975), the EVT requires choosing all the values over a given threshold,us, and not only some of them, as in the case of physical declustering using the peaks over up.”
Extended generalised Pareto models for tail estimation (2011, background): “ in (0;1) (1 ;1) and z + = max(z;0). Consider a random variable Xarising from an absolutely continuous distribution function F X and let also xF X = supfx: F X(x) <1gbe the upper end point of F X. Pickands (1975) shows that if there exists a scaling function h X(u) : R !R +, u<xF X, such that ”
GENERALIZED DOUBLE PARETO SHRINKAGE. (2011, unknown): “In contrast to (2.1), the generalized Pareto density of Pickands (1975) is parametrized in terms of a location parameter µ ∈ R, a scale parameter ξ > 0, and a shape parameter α ∈ R as f(θ | ξ, α, µ) = 1 ξ ( 1 + θ − µ αξ )−(α+1) , (2.2) with θ ≥ µ for α > 0 and µ ≤ θ ≤ µ − ξα for α 0.”
Estimation of Hüsler–Reiss distributions and Brown–Resnick processes (2012, background): “…suitable Poisson point process to all data that exceed a certain high threshold and thus follow approximately a generalized Pareto distribution (see Pickands (1975)), i.e. for y> 0 P.X>y +v|X>v/≈ { .1+ ζy=s/−1=ζ+ , ζ =0, exp.−y=s/, ζ =0, .1/ where a+ =max.0, a/, ζ and s> 0 are the shape and scale…”