How to make a Hill plot

Drees, Holger; Haan, L.F.M. de; Resnick, Sidney I.

doi:10.1214/aos/1016120372

Cited by 193 publications

(129 citation statements)

References 22 publications

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“…As the order statistics sample path U n (·) has the same distribution as the self-normalized sample path N n (·), the sample paths of the order statistics of uniformly distributed random variables {U n (·)} satisfy the LDP in V + 0,1 with the rate function given in equation (8). As log(0) = −∞, J U (ψ) = ∞ unless the absolutely continuous component of ψ's Lebesgue decomposition is strictly increasing almost everywhere with respect to Lebesgue measure.…”

Section: Functional Ldp For Order Statisticsmentioning

confidence: 99%

Sample Path Large Deviations for Order Statistics

Duffy

Macci

Torrisi

2011

Journal of Applied Probability

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We consider the sample paths of the order statistics of i.i.d. random variables with common distribution function F . If F is strictly increasing but possibly having discontinuities, we prove that the sample paths of the order statistics satisfy the large deviation principle in the Skorohod M 1 topology. Sanov's Theorem is deduced in the Skorohod M 1 topology as a corollary to this result. A number of illustrative examples are presented, including applications to the sample paths of trimmed means and Hill Plots.

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Section: Functional Ldp For Order Statisticsmentioning

confidence: 99%

Sample Path Large Deviations for Order Statistics

Duffy

Macci

Torrisi

2011

Journal of Applied Probability

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“…Both plots are consistent with D and R being heavy tailed with tail parameters α D and α R greater than 1. (This is confirmed [12,26,29] that the Hillish plots of (A, θ 2 ) and (A, −θ 2 ) are close to 1 near the left side of their plots. We also observe that the Pickandsish estimate at q = 0.8 remains near 0.…”

Section: Testing For Mrv and Hrv: Data Examplesmentioning

confidence: 59%

Models with Hidden Regular Variation: Generation and Detection

Das

Resnick

2015

Stochastic Systems

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We review the notions of multivariate regular variation (MRV) and hidden regular variation (HRV) for distributions of random vectors and then discuss methods for generating models exhibiting both properties concentrating on the non-negative orthant in dimension two. Furthermore we suggest diagnostic techniques that detect these properties in multivariate data and indicate when models exhibiting both MRV and HRV are plausible fits for the data. We illustrate our techniques on simulated data, as well as two real Internet data sets.1. Introduction. This paper discusses methods for constructing multivariate heavy-tailed models on particular sub-cones of R 2 + = [0, ∞) 2 by adapting techniques from multivariate regular variation theory. We evaluate two different methods for generating models exhibiting multiple heavy-tailed regimes. The results obtained in the different regimes are governed by the sub-cone that serves as the state-space, the choice of scaling function, and often the interaction between different regimes. We also discuss statistical detection methods which validate that data is consistent with particular multivariate heavy-tailed models; these methods are adapted from those developed for the conditional extreme value model (CEV) [8]. We discuss multivariate regular variation (MRV) on the cones R 2 + \ {0} and (0, ∞) 2 . When regular variation exists on both cones, the regular variation on the smaller cone (0, ∞) 2 is called hidden regular variation (HRV).Data that may be modeled by distributions having heavy tails appear in many contexts, for example, hydrology [1], finance [31], insurance [14], Internet traffic [6], social networks and random graphs [3,13,28,30] and risk

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“…It is customary to estimate γ * value through a graphical representation called a Hill plot. This type of depiction is formed by plotting γ * versus k [71]. For EV distributions (e.g., Pareto), estimated γ * (or, related p max via Eq.…”

Section: Estimating P Max From Limited Datamentioning

confidence: 99%

Estimating higher-order structure functions from geophysical turbulence time series: Confronting the curse of the limited sample size

DeMarco

Basu

2017

Phys. Rev. E

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Utilizing synthetically generated random variates and laboratory measurements, we document the inherent limitations of the conventional structure function approach in limited sample size settings. We demonstrate that an alternative approach, based on the principle of maximum likelihood, can provide nearly unbiased structure function estimates with far less uncertainty under such unfavorable conditions. The superiority of this approach over the conventional approach does not diminish even in the case of strongly correlated samples. Two entirely different types of probability distributions, which have been reported in the turbulence literature, are found to be compatible with the proposed approach.

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How to make a Hill plot

Cited by 193 publications

References 22 publications

Sample Path Large Deviations for Order Statistics

Sample Path Large Deviations for Order Statistics

Models with Hidden Regular Variation: Generation and Detection

Estimating higher-order structure functions from geophysical turbulence time series: Confronting the curse of the limited sample size

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