Univariate Extreme Value Analysis

Dey, Dipak K.; Roy, Dooti; Yan, Jun

doi:10.1201/b19721-5

Cited by 25 publications

(22 citation statements)

References 21 publications

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“…Moreover, if we assume A ( t )= ξβt ρ , ρ <0 and ω a fixed value, the threshold k 0 that minimizes the AMSE, the so‐called optimal level for the estimation of ξ (Brilhante et al, Hall, and Caeiro and Gomes, among others), is given by

k_{0}^{•} = k_{0}^{•} (n) = \arg \min_{k} A M S E ({\hat{ξ}}_{n}^{•} (k)) = {(\frac{σ_{•}^{2}}{(- 2 ρ) b_{•}^{2} ξ^{2} β^{2}})}^{\frac{1}{1 - 2 ρ}} n^{\frac{- 2 ρ}{1 - 2 ρ}} (1 + o (1)) .

For the estimators under consideration, it should be noted that

σ_{•}^{2} false/ ξ^{2}

, and consequently,

k_{0}^{•}

are independent of ξ .…”

Section: Results For the Classes Of Kernel Evi Estimatorsmentioning

confidence: 99%

“…The estimation of β has been performed through the use of a statistics dependent on a consistent estimator of ρ , denoted

\overset{β}{true} false(k; \overset{ρ}{true} false)

, introduced in Gomes and Martins, and also computed at the same k 1 , ie,

\overset{β}{true} = \overset{β}{true} false(k_{1}; \overset{ρ}{true} false)

. Algorithms for the estimation of the second‐order parameters ( ρ , β ) can be found in Gomes and Pestana, Gomes et al, and Caeiro and Gomes, among others.…”

Section: New Classes Of Reduced Bias Kernel Estimatorsmentioning

confidence: 99%

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Reduced‐bias kernel estimators of a positive extreme value index

Caeiro

Henriques‐Rodrigues

2019

Math Methods in App Sciences

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In this paper, we deal with the semi‐parametric estimation of the extreme value index, an important parameter in extreme value analysis. It is well known that many classic estimators, such as the Hill estimator, reveal a strong bias. This problem motivated the study of two classes of kernel estimators. Those classes generalize the classical Hill estimator and have a tuning parameter that enables us to modify the asymptotic mean squared error and eventually to improve their efficiency. Since the improvement in efficiency is not very expressive, we also study new reduced bias estimators based on the two classes of kernel statistics. Under suitable conditions, we prove their asymptotic normality. Moreover, an asymptotic comparison, at optimal levels, shows that the new classes of reduced bias estimators are more efficient than other reduced bias estimator from the literature. An illustration of the finite sample behaviour of the kernel reduced‐bias estimators is also provided through the analysis of a data set in the field of insurance.

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k_{0}^{•} = k_{0}^{•} (n) = \arg \min_{k} A M S E ({\hat{ξ}}_{n}^{•} (k)) = {(\frac{σ_{•}^{2}}{(- 2 ρ) b_{•}^{2} ξ^{2} β^{2}})}^{\frac{1}{1 - 2 ρ}} n^{\frac{- 2 ρ}{1 - 2 ρ}} (1 + o (1)) .

For the estimators under consideration, it should be noted that

σ_{•}^{2} false/ ξ^{2}

, and consequently,

k_{0}^{•}

are independent of ξ .…”

Section: Results For the Classes Of Kernel Evi Estimatorsmentioning

confidence: 99%

“…The estimation of β has been performed through the use of a statistics dependent on a consistent estimator of ρ , denoted

\overset{β}{true} false(k; \overset{ρ}{true} false)

, introduced in Gomes and Martins, and also computed at the same k 1 , ie,

\overset{β}{true} = \overset{β}{true} false(k_{1}; \overset{ρ}{true} false)

. Algorithms for the estimation of the second‐order parameters ( ρ , β ) can be found in Gomes and Pestana, Gomes et al, and Caeiro and Gomes, among others.…”

Section: New Classes Of Reduced Bias Kernel Estimatorsmentioning

confidence: 99%

Reduced‐bias kernel estimators of a positive extreme value index

Caeiro

Henriques‐Rodrigues

2019

Math Methods in App Sciences

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“…There exist some methods for choosing the thresholdtype tuning parameter; see Caeiro and Gomes (2016) for a review on this topic. In practice, we choose k 0 by adapting the procedure in Neves et al (2015) based on the path stability.…”

Section: Simulation Studymentioning

confidence: 99%

Extremal linear quantile regression with Weibull-type tails

Wang

Tong³

2020

STAT SINICA

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This paper studies the estimation of extreme conditional quantiles for distributions with Weibull-type tails. We propose two families of estimators for the Weibull tail-coefficient, and construct an extrapolation estimator for the extreme conditional quantiles based on quantile regression and extreme value theory. The asymptotic results of the proposed estimators are established. The work fills a gap in the literature of extreme quantile regression where many important Weibull-type distributions are excluded by the assumed strong conditions. We demonstrate through simulation study that the proposed Statistica Sinica: Newly accepted Paper (accepted author-version subject to English editing) Extremal linear quantile regression with Weibull-type tails extrapolation method provides more efficient and stable estimation of conditional quantiles of extreme orders than the conventional method.The practical value of the proposed method is demonstrated through the analysis of extremely high birth weight.

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“…, d, and the dependence structure of Y −u | Y ≤ u should be well-approximated by that of a multivariate GP distribution. Marginal threshold selection has a large body of literature devoted to it; see Scarrott and MacDonald (2012) and Caeiro and Gomes (2016) for recent reviews. Threshold selection for dependence models is a much less well studied problem.…”

Section: Threshold Selection and Model Diagnosticsmentioning

confidence: 99%

Peaks Over Thresholds Modeling With Multivariate Generalized Pareto Distributions

et al. 2018

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When assessing the impact of extreme events, it is often not just a single component, but the combined behaviour of several components which is important. Statistical modelling using multivariate generalized Pareto (GP) distributions constitutes the multivariate analogue of univariate peaks over thresholds modelling, which is widely used in finance and engineering. We develop general methods for construction of multivariate GP distributions and use them to create a variety of new statistical models. A censored likelihood procedure is proposed to make inference on these models, together with a threshold selection procedure, goodness-of-fit diagnostics, and a computationally tractable strategy for model selection. The models are fitted to returns of stock prices of four UK-based banks and to rainfall data in the context of landslide risk estimation. Supplementary materials and codes are available online.

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Univariate Extreme Value Analysis

Cited by 25 publications

References 21 publications

Reduced‐bias kernel estimators of a positive extreme value index

Reduced‐bias kernel estimators of a positive extreme value index

Extremal linear quantile regression with Weibull-type tails

Peaks Over Thresholds Modeling With Multivariate Generalized Pareto Distributions

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