Extreme-Value Analysis of Standardized Gaussian Increments

Kabluchko, Zakhar

doi:10.48550/arxiv.0706.1849

Cited by 5 publications

(7 citation statements)

References 28 publications

(41 reference statements)

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“…, there exist constants a n > 0 and b n ∈ R such that (a −1 n (M n − b n )) has a Gumbel limit distribution. Another proof of this result is given in [16]. Finally, the famous Erdős-Rényi laws are also closely related to the maximum increments of a random walk.…”

Section: Introductionmentioning

confidence: 85%

The limit distribution of the maximum increment of a random walk with regularly varying jump size distribution

Mikosch

Račkauskas

2010

Bernoulli

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In this paper, we deal with the asymptotic distribution of the maximum increment of a random walk with a regularly varying jump size distribution. This problem is motivated by a longstanding problem on change point detection for epidemic alternatives. It turns out that the limit distribution of the maximum increment of the random walk is one of the classical extreme value distributions, the Fréchet distribution. We prove the results in the general framework of point processes and for jump sizes taking values in a separable Banach space.Keywords: Banach space valued random element; epidemic change point; extreme value theory; Fréchet distribution; maximum increment of a random walk; point process convergence; regular variation This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2010, Vol. 16, No. 4, 1016-1038. This reprint differs from the original in pagination and typographic detail. 1350-7265 c 2010 ISI/BSThe limit distribution of the maximum increment of a random walk 1017 mention that nothing seems to be known about the distributional properties of T n . There exist several approaches to replace the original problem by a more tractable one. One way is to restrict the range over which the maximum is taken to ℓ n ≤ ℓ ≤ n − ℓ n for some ℓ n → ∞ satisfying ℓ n = o(n); see, for example, Yao [24]. Alternatively, one can change the normalizing constants ℓ(1 − ℓ/n) in a suitable way; see, for example, Račkauskas and Suquet [20].Statistics of type T n appear in the context of tests for change points in the mean under epidemic alternatives. This problem can be formulated as follows: given that X 1 , . . . , X n are independent random variables, test the null hypothesis of constant mean • H 0 : EX 1 = EX 2 = · · · = EX n = µ against the epidemic alternative • H A : There exist integers 1 ≤ k * < m * < n such that EX 1 = · · · = EX k * = EX m * +1 = · · · = EX n = µ, EX k * +1 = · · · = EX m * = ν and µ = ν.

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Section: Introductionmentioning

confidence: 85%

The limit distribution of the maximum increment of a random walk with regularly varying jump size distribution

Mikosch

Račkauskas

2010

Bernoulli

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“…This guarantees a coverage probability of at least α = 0.95 for all samples of size n ≥ 500 and it tends rapidly to one as the sample size increases. The exact asymptotic distribution of max 1≤i<j≤n ( j l=i Z l ) 2 /(j − i + 1) has recently been derived by Kabluchko (2007).…”

Section: 2mentioning

confidence: 99%

Nonparametric regression, confidence regions and regularization

Davies¹,

Kovac²,

Meise³

2009

Ann. Statist.

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In this paper we offer a unified approach to the problem of nonparametric regression on the unit interval. It is based on a universal, honest and non-asymptotic confidence region An which is defined by a set of linear inequalities involving the values of the functions at the design points. Interest will typically centre on certain simplest functions in An where simplicity can be defined in terms of shape (number of local extremes, intervals of convexity/concavity) or smoothness (bounds on derivatives) or a combination of both. Once some form of regularization has been decided upon the confidence region can be used to provide honest non-asymptotic confidence bounds which are less informative but conceptually much simpler.

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“…These show that for I n = I n (2) we have τ n (0.95) ≤ 3 for all n ≥ 500. If I n contains all singletons {t i }, as will always be the case, it follows from Dümbgen and Spokoiny (2001) and Kabluchko (2007) that lim n→∞ τ n (α) = 2 for any α. One immediate consequence of ( 11) is…”

Section: Single Samplesmentioning

confidence: 87%

Quantifying the cost of simultaneous non-parametric approximation of several samples

Davies¹,

Kovac²

2009

Electron. J. Statist.

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We consider the standard non-parametric regression model with Gaussian errors but where the data consist of different samples. The question to be answered is whether the samples can be adequately represented by the same regression function. To do this we define for each sample a universal, honest and non-asymptotic confidence region for the regression function. Any subset of the samples can be represented by the same function if and only if the intersection of the corresponding confidence regions is non-empty. If the empirical supports of the samples are disjoint then the intersection of the confidence regions is always non-empty and a negative answer can only be obtained by placing shape or quantitative smoothness conditions on the joint approximation. Alternatively a simplest joint approximation function can be calculated which gives a measure of the cost of the joint approximation, for example, the number of extra peaks required.AMS 2000 subject classifications: Primary 62G08; secondary 62G15, 62P35, 82D25.

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Extreme-Value Analysis of Standardized Gaussian Increments

Cited by 5 publications

References 28 publications

The limit distribution of the maximum increment of a random walk with regularly varying jump size distribution

The limit distribution of the maximum increment of a random walk with regularly varying jump size distribution

Nonparametric regression, confidence regions and regularization

Quantifying the cost of simultaneous non-parametric approximation of several samples

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