Non-central limit theorems for random selections

Berkes, István; Horváth, Lajos; Schauer, Johannes

doi:10.1007/s00440-009-0212-z

Cited by 9 publications

(7 citation statements)

References 14 publications

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“…So, resampling cannot be recommended to obtain asymptotic critical values. This result was obtained by Aue et al [2] for permutation resampling and by Athreya [1], Hall [18] and Berkes et al [4] for the bootstrap. No efficient procedure has been found to test H 0 against H A when EX 2 j = ∞.…”

Section: Asymptotics Of Trimmed Cusum Statisticssupporting

confidence: 58%

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Asymptotics of trimmed CUSUM statistics

Berkes

Horváth²,

Schauer³

2011

Bernoulli

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There is a wide literature on change point tests, but the case of variables with infinite variances is essentially unexplored. In this paper we address this problem by studying the asymptotic behavior of trimmed CUSUM statistics. We show that in a location model with i.i.d. errors in the domain of attraction of a stable law of parameter $0<\alpha <2$, the appropriately trimmed CUSUM process converges weakly to a Brownian bridge. Thus, after moderate trimming, the classical method for detecting change points remains valid also for populations with infinite variance. We note that according to the classical theory, the partial sums of trimmed variables are generally not asymptotically normal and using random centering in the test statistics is crucial in the infinite variance case. We also show that the partial sums of truncated and trimmed random variables have different asymptotic behavior. Finally, we discuss resampling procedures which enable one to determine critical values in the case of small and moderate sample sizes.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ318 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

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Section: Asymptotics Of Trimmed Cusum Statisticssupporting

confidence: 58%

“…By the results of Aue et al [2] and Berkes et al [4], if we sample from the original (untrimmed) observations, then the CUSUM process converges weakly to a non-Gaussian process containing random parameters and thus the resampling procedure is statistically useless.…”

Section: Resampling Methodsmentioning

confidence: 99%

Asymptotics of trimmed CUSUM statistics

Berkes

Horváth²,

Schauer³

2011

Bernoulli

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“…Csörgö and Dodunekova [3] showed that merging holds for extremal and trimmed sums of the sequence (X n ) as well and Berkes et al [4] and del Barrio et al [5] proved that the same holds for bootstrapped sums of (X n ). Let Z(t) denote the Lévy process defined by…”

Section: Introductionmentioning

confidence: 87%

Strong approximation of the St. Petersburg game

Berkes

2017

Statistics

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Let X, X 1 , X 2 , . . . be i.i.d. random variables with P(X = 2 k ) = 2 −k (k ∈ N) and let S n = n k=1 X k . The properties of the sequence S n have received considerable attention in the literature in connection with the St. Petersburg paradox (Bernoulli 1738). Let {Z(t), t ≥ 0} be a semistable Lévy process with underlying Lévy measure k∈Z 2 −k δ 2 k . For a suitable version of (X k ) and Z(t), we prove the strong approximation S n = Z(n) + O(n 5/6+ε ) a.s. This provides the first example for a strong approximation theorem for partial sums of i.i.d. sequences not belonging to the domain of attraction of the normal or stable laws. ARTICLE HISTORY

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“…In Aue et al [2] it was shown that U n (t), suitably normalized, converges weakly to a non-Gaussian process, whose distribution, however, is not known explicitly, and thus critical values for the test cannot be computed analytically. A natural solution of this difficulty is to use a bootstrap or permutation test, but, as it was shown in [2] and [3], the normalized CUSUM functional converges in this case to a continuous process containing random parameters 1150002-11 Stoch. Dyn.…”

Section: Theorem 33 Let As X → ∞mentioning

confidence: 99%

Asymptotic Behavior of Trimmed Sums

2012

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Trimming is a standard method to decrease the effect of large sample elements in statistical procedures, used, e.g., for constructing robust estimators. It is also a powerful tool in understanding deeper properties of partial sums of independent random variables. In this paper we review some basic results of the theory and discuss new results in the central limit theory of trimmed sums. In particular, we show that for random variables in the domain of attraction of a stable law with parameter 0 < α < 2, the asymptotic behavior of modulus trimmed sums depends sensitively on the number of elements eliminated from the sample. We also show that under moderate trimming, the central limit theorem always holds if we allow random centering factors. Finally, we give an application to change point problems.

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Non-central limit theorems for random selections

Cited by 9 publications

References 14 publications

Asymptotics of trimmed CUSUM statistics

Asymptotics of trimmed CUSUM statistics

Strong approximation of the St. Petersburg game

Asymptotic Behavior of Trimmed Sums

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