Linear Functions of Uniform Order Statistics and B-Splines

Agarwal, Gaurav; Dalpatadu, Rohan J.; Singh, Ashok K.

doi:10.1081/sta-120002644

Cited by 11 publications

(15 citation statements)

References 25 publications

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“…, d}, we defineĊ j to be zero on the set {u ∈ [0, 1] d : u j ∈ {0, 1}} (see also Segers [43]; Bücher and Volgushev [13]). It then follows that, under Condition 3.2,Ċ j is defined on the whole of [0,1] d . Also, for any j ∈ {1, .…”

Section: A Dependent Multiplier Bootstrap For the Multivariate Empirimentioning

confidence: 99%

“…Notice that κ T = κ U,1 , κ B = κ U,2 and κ P = κ U,4 . This also implies that κ U,8 is a rescaled and normalized version of the convolution of κ P with itself, that is, κ U,8 (x) = κ P ⋆ κ P (2x)/κ P ⋆ κ P (0) for all x ∈ R. A numerically stable and efficient way of computing κ U,p consists of using divided differences (see, e.g., Agarwal, Dalpatadu and Singh [1]). Finally, note that the truncated and flat top kernels cannot be used as they do not satisfy the integral condition ensuring that Σ n is positive definite.…”

Section: The Covariance Matrix Approachmentioning

confidence: 99%

“…The underlying idea is as follows: the fact that the limiting vector of processes in Theorem 2.1 has independent components suggests regardingB (1) n , . .…”

Section: A Dependent Multiplier Bootstrap For C N Under Strong Mixingmentioning

confidence: 99%

“…. , X n later and fix u, v ∈ [0, 1] d . Proceeding in the spirit of Politis and Romano [37] and Politis [36], Figure 1.…”

Section: Estimation Of the Bandwidth Parameter ℓ Nmentioning

confidence: 99%

“…C, and let µ be a measure on [0,1] d+1 defined by µ = 2λ⊗ ν, where λ denotes the one-dimensional Lebesgue measure. Next, for some real q ∈ (2a/(a − 3), 6a…”

Section: Monte Carlo Experimentsmentioning

confidence: 99%

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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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Section: A Dependent Multiplier Bootstrap For the Multivariate Empirimentioning

confidence: 99%

Section: The Covariance Matrix Approachmentioning

confidence: 99%

“…The underlying idea is as follows: the fact that the limiting vector of processes in Theorem 2.1 has independent components suggests regardingB (1) n , . .…”

Section: A Dependent Multiplier Bootstrap For C N Under Strong Mixingmentioning

confidence: 99%

“…. , X n later and fix u, v ∈ [0, 1] d . Proceeding in the spirit of Politis and Romano [37] and Politis [36], Figure 1.…”

Section: Estimation Of the Bandwidth Parameter ℓ Nmentioning

confidence: 99%

“…C, and let µ be a measure on [0,1] d+1 defined by µ = 2λ⊗ ν, where λ denotes the one-dimensional Lebesgue measure. Next, for some real q ∈ (2a/(a − 3), 6a…”

Section: Monte Carlo Experimentsmentioning

confidence: 99%

See 3 more Smart Citations