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

Abstract

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.

Nonparametric inference of gradual changes in the jump behaviour of time-continuous processes

“…As a consequence (4.9) follows from (6.34) and Proposition F.1 in the supplement to Bücher and Kojadinovic (2016) as soon as we can show…”

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

“…, S (B) n , respectively. The following theorem, whose proof can be found in Section F of the the supplement to Bücher and Kojadinovic (2013), shows that the tests which reject H 0 for T n > t n,B,1−α or for S n > s n,B,1−α asymptotically hold their level, for B → ∞ and n → ∞. …”

“…data sets, and has been applied to different testing issues in Genest et al (2011);Bücher et al (2011Bücher et al ( , 2012, among others. The procedure has been generalized to the alpha-mixing case in Bücher and Ruppert (2013) with several further improvements in Bücher and Kojadinovic (2013).…”

“…, C (B) n are 'almost' independent copies of C n and can hence be used to approximate its distribution. For a proof, see Bücher and Kojadinovic (2013) for the alpha-mixing case or Segers (2012) for the i.i.d. case.…”

“…We prove weak convergence of the corresponding process √ n(C n − C θn ). In order to get access to the critical values of the test statistcs, we adapt the dependent multiplier bootstrap of Bücher and Kojadinovic (2013) to the present setting, following ideas of and . The finite-sample performance of our methods are investigated by Monte Carlo experiments.…”