Empirical Process Techniques for Dependent Data

Dehling, Herold; Philipp, Walter

doi:10.1007/978-1-4612-0099-4_1

Cited by 101 publications

(121 citation statements)

References 106 publications

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“…On the other hand, by Lemma 3.11 of Dehling and Philipp [20], for any q ∈ (2a/(a − 3), 4) and p ∈ (2, a/3) such that 1/p + 2/q = 1, we have…”

Section: Monte Carlo Experimentsmentioning

confidence: 95%

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A dependent multiplier bootstrap for the sequential empirical copula process under strong mixing

Bücher¹,

Kojadinovic²

2016

Bernoulli

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Two key ingredients to carry out inference on the copula of multivariate observations are the empirical copula process and an appropriate resampling scheme for the latter. Among the existing techniques used for i.i.d. observations, the multiplier bootstrap of Rémillard and Scaillet (J. Multivariate Anal. 100 (2009) 377-386) frequently appears to lead to inference procedures with the best finite-sample properties. Bücher and Ruppert (J. Multivariate Anal. 116 (2013) 208-229) recently proposed an extension of this technique to strictly stationary strongly mixing observations by adapting the dependent multiplier bootstrap of Bühlmann (The blockwise bootstrap in time series and empirical processes (1993) ETH Zürich, Section 3.3) to the empirical copula process. The main contribution of this work is a generalization of the multiplier resampling scheme proposed by Bücher and Ruppert along two directions. First, the resampling scheme is now genuinely sequential, thereby allowing to transpose to the strongly mixing setting many of the existing multiplier tests on the unknown copula, including nonparametric tests for change-point detection. Second, the resampling scheme is now fully automatic as a data-adaptive procedure is proposed which can be used to estimate the bandwidth parameter. A simulation study is used to investigate the finite-sample performance of the resampling scheme and provides suggestions on how to choose several additional parameters. As by-products of this work, the validity of a sequential version of the dependent multiplier bootstrap for empirical processes of Bühlmann is obtained under weaker conditions on the strong mixing coefficients and the multipliers, and the weak convergence of the sequential empirical copula process is established under many serial dependence conditions. Keywords: lag window estimator; multiplier central limit theorem; multivariate observations; partial-sum process; ranks; serial dependence This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2016, Vol. 22, No. 2, 927-968. This reprint differs from the original in pagination and typographic detail.

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“…On the other hand, by Lemma 3.11 of Dehling and Philipp [20], for any q ∈ (2a/(a − 3), 4) and p ∈ (2, a/3) such that 1/p + 2/q = 1, we have…”

Section: Monte Carlo Experimentsmentioning

confidence: 95%

“…The following result can be proved using a well-known blocking technique (see, e.g., Dehling and Philipp [20], page 31). Its proof is given in the supplementary material (Bücher and Kojadinovic [9]).…”

Section: Monte Carlo Experimentsmentioning

confidence: 98%

A dependent multiplier bootstrap for the sequential empirical copula process under strong mixing

Bücher¹,

Kojadinovic²

2016

Bernoulli

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“…For a countable class with a finite VC dimension, a GC theorem has been recently shown by Adams and Nobel [2] for ergodic sequences. We refer the reader to [2,10] for a more comprehensive survey of results for non-i.i.d. data.…”

Section: Related Literaturementioning

confidence: 99%

Sequential complexities and uniform martingale laws of large numbers

Rakhlin

Sridharan

Tewari

2014

Probab. Theory Relat. Fields

122

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We establish necessary and sufficient conditions for a uniform martingale Law of Large Numbers. We extend the technique of symmetrization to the case of dependent random variables and provide "sequential" (noni.i.d.) analogues of various classical measures of complexity, such as covering numbers and combinatorial dimensions from empirical process theory. We establish relationships between these various sequential complexity measures and show that they provide a tight control on the uniform convergence rates for empirical processes with dependent data. As a direct application of our results, we provide exponential inequalities for sums of martingale differences in Banach spaces.Keywords empirical processes, dependent data, uniform Glivenko-Cantelli classes, rademacher averages, sequential prediction Disciplines Statistics and ProbabilityThis journal article is available at ScholarlyCommons: http://repository.upenn.edu/statistics_papers/531 Sequential Complexities and Uniform Martingale Laws of Large NumbersAlexander Rakhlin Karthik Sridharan Ambuj TewariSeptember 19, 2013 AbstractWe establish necessary and sufficient conditions for a uniform martingale Law of Large Numbers. We extend the technique of symmetrization to the case of dependent random variables and provide "sequential" (non-i.i.d.) analogues of various classical measures of complexity, such as covering numbers and combinatorial dimensions from empirical process theory. We establish relationships between these various sequential complexity measures and show that they provide a tight control on the uniform convergence rates for empirical processes with dependent data. As a direct application of our results, we provide exponential inequalities for sums of martingale differences in Banach spaces.

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“…To this end, we will frequently exploit the following bounds which are consequences of Lemma 3.9 and Lemma 3.11 in Dehling and Philipp (2002) …”

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confidence: 99%

“…The fact that |E[Z (b) i,n Z (b) j,n ]| is bounded by 1 together with an application of Lemma 3.11 in Dehling and Philipp (2002) with r = s = p and t = p/(p − 2) allows to bound the right-hand side of the last display by 10 n n i,j=1…”

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confidence: 99%

Goodness-of-Fit Tests for Multivariate Copula-Based Time Series Models

Berghaus

Bücher

2016

Econom. Theory

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In recent years, stationary time series models based on copula functions became increasingly popular in econometrics to model nonlinear temporal and cross-sectional dependencies. Within these models, we consider the problem of testing the goodness-of-fit of the parametric form of the underlying copula. Our approach is based on a dependent multiplier bootstrap and it can be applied to any stationary, strongly mixing time series. The method extends recent i.i.d. results by Kojadinovic, Yan and Holmes [I. Kojadinovic, Y. Yan and M. Holmes, Fast large sample goodness-of-fit tests for copulas, Statistica Sinica 21 (2011), 841-871] and shares the same computational benefits compared to methods based on a parametric bootstrap. The finite-sample performance of our approach is investigated by Monte Carlo experiments for the case of copula-based Markovian time series models.

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Empirical Process Techniques for Dependent Data

Cited by 101 publications

References 106 publications

A dependent multiplier bootstrap for the sequential empirical copula process under strong mixing

A dependent multiplier bootstrap for the sequential empirical copula process under strong mixing

Sequential complexities and uniform martingale laws of large numbers

Goodness-of-Fit Tests for Multivariate Copula-Based Time Series Models

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