On bootstrap estimation of the distribution of the studentized mean

Hall, Peter; LePage, Raoul

doi:10.1007/bf00050845

Cited by 22 publications

(17 citation statements)

References 13 publications

(8 reference statements)

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“…Our approach can be employed to construct consistent confidence regions even when the error distribution does not lie in any domain of attraction. Compare, for example, Hall and LePage (1996). However, on the present occasion such a degree of generality would be a significant distraction, and we do not pursue it.…”

Section: Confidence Regionsmentioning

confidence: 87%

Inference in Arch and Garch Models with Heavy-Tailed Errors

2003

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ARCH and GARCH models directly address the dependency of conditional second moments, and have proved particularly valuable in modelling processes where a relatively large degree of fluctuation is present. These include financial time series, which can be particularly heavy tailed. However, little is known about properties of ARCH or GARCH models in the heavy–tailed setting, and no methods are available for approximating the distributions of parameter estimators there. In this paper we show that, for heavy–tailed errors, the asymptotic distributions of quasi–maximum likelihood parameter estimators in ARCH and GARCH models are nonnormal, and are particularly difficult to estimate directly using standard parametric methods. Standard bootstrap methods also fail to produce consistent estimators. To overcome these problems we develop percentile–t, subsample bootstrap approximations to estimator distributions. Studentizing is employed to approximate scale, and the subsample bootstrap is used to estimate shape. The good performance of this approach is demonstrated both theoretically and numerically.

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Section: Confidence Regionsmentioning

confidence: 87%

Inference in Arch and Garch Models with Heavy-Tailed Errors

2003

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“…which has asymptotic coverage probability 1 − α under very mild regularity conditions, including n 1 → ∞ and n 1 /n → 0 as n → ∞ [see Hall and LePage (1996) for details]. We generated 500 pseudorandom samples of size n = 1000 from one of the two distributions:…”

Section: Empirical Likelihood With Heavy Tailsmentioning

confidence: 99%

Empirical-likelihood-based confidence interval for the mean with a heavy-tailed distribution

Peng

2004

Ann. Statist.

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Empirical-likelihood-based confidence intervals for a mean were introduced by Owen [Biometrika 75 (1988) 237-249], where at least a finite second moment is required. This excludes some important distributions, for example, those in the domain of attraction of a stable law with index between 1 and 2. In this article we use a method similar to Qin and Wong [Scand. J. Statist. 23 (1996) 209-219] to derive an empirical-likelihood-based confidence interval for the mean when the underlying distribution has heavy tails. Our method can easily be extended to obtain a confidence interval for any order of moment of a heavy-tailed distribution.

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“…With regard to the traditional bootstrap in the context of regression with long tailed errors the work of Arcones and Gine (1989) proves that taking smaller resample size of the order o(n) nÂlog log n allows recovery of the asymptotic unconditional distribution of estimation errors for the sample mean case (see also Hall and LePage, 1995). This approach, if it could be applied to least squares, would enable one to develop confidence intervals based on the unconditional asymptotic sampling distribution of the least squares estimators.…”

Section: Introductionmentioning

confidence: 99%

Resampling Permutations in Regression without Second Moments

LePage

Podgórski²

1996

Journal of Multivariate Analysis

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For given observations we consider the resampling distribution obtained by permuting residuals versus the distribution of errors conditional on their order statistics. We observe that with high probability both are approximately equal on contrast vectors after suitable normalization. No momentÂdistribution assumptions on the underlying errors other than exchangeability are required. These results remain valid if we consider reduced regression models derived from the original one by removing data for which the corresponding residuals take extreme values. We obtain general conditions under which confidence regions produced by such methods are accurate. In the absence of finite second moments randomness of the limiting bootstrap distributions has been considered as a failure of the traditional bootstrap and no practical meaning has been given to the phenomenon. For our method this type of randomness is still present in the limit but it has clear probabilistic interpretation as a conditional distribution which can be used, e.g., to obtain conditional confidence sets. We study this phenomenon in detail for the case of independent errors with distribution from the domain of attraction of stable laws.

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On bootstrap estimation of the distribution of the studentized mean

Abstract: Bootstrap, central limit theorem, consistency, domain of attraction, domain of partial attraction, heavy tail, percentile-t method, self-normalization, Stable law, Studentization,

Cited by 22 publications

References 13 publications

Inference in Arch and Garch Models with Heavy-Tailed Errors

Inference in Arch and Garch Models with Heavy-Tailed Errors

Empirical-likelihood-based confidence interval for the mean with a heavy-tailed distribution

Resampling Permutations in Regression without Second Moments

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