Weak convergence of the empirical copula process with respect to weighted metrics

Berghaus, Betina; Bücher, Axel; Volgushev, Stanislav

doi:10.3150/15-bej751

Cited by 30 publications

(86 citation statements)

References 41 publications

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“…Empirical copula processes were recently shown to converge weakly also with respect to stronger metrics than the supremum distance. A first seminal result in that direction is due to Berghaus et al [1] for the empirical copula processes C n andC n . Berghaus and Segers [2] have shown a similar result for the empirical beta copula process C β n .…”

Section: Subsampling Weighted Empirical Copula Processesmentioning

confidence: 99%

Subsampling (weighted smooth) empirical copula processes

Kojadinovic

Stemikovskaya

2019

Journal of Multivariate Analysis

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A key tool to carry out inference on the unknown copula when modeling a continuous multivariate distribution is a nonparametric estimator known as the empirical copula. One popular way of approximating its sampling distribution consists of using the multiplier bootstrap. The latter is however characterized by a high implementation cost. Given the rank-based nature of the empirical copula, the classical empirical bootstrap of Efron does not appear to be a natural alternative, as it relies on resamples which contain ties. The aim of this work is to investigate the use of subsampling in the aforementioned framework. The latter consists of basing the inference on statistic values computed from subsamples of the initial data. One of its advantages in the rank-based context under consideration is that the formed subsamples do not contain ties. Another advantage is its asymptotic validity under minimalistic conditions. In this work, we show the asymptotic validity of subsampling for several (weighted, smooth) empirical copula processes both in the case of serially independent observations and time series. In the former case, subsampling is observed to be substantially better than the empirical bootstrap and equivalent, overall, to the multiplier bootstrap in terms of finite-sample performance.

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Section: Subsampling Weighted Empirical Copula Processesmentioning

confidence: 99%

Subsampling (weighted smooth) empirical copula processes

Kojadinovic

Stemikovskaya

2019

Journal of Multivariate Analysis

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“…Denote first ( ) = 1 ∑ =1 ( ( ) ⩽ ). Then, it follows from proposition 4.4 of Berghaus et al (2017) that for any ∈ (0, 1∕2), ∈ (0, 1) and → 0,…”

Section: Proofsmentioning

confidence: 98%

“…A key technique in deriving our theoretical results is the following weighted approximation of empirical copula process: Under the regularity condition C1) given in the next paragraph, it follows from proposition 4.4 of Berghaus, Bücher, and Volgushev () that

\underset{0 ⩽ x_{1}, \dots, x_{d - 1}, y ⩽ 1}{trueprefixsup} false| \sqrt{n} true(C_{n} (x_{1}, \dots, x_{d - 1}, y) - C (x_{1}, \dots, x_{d - 1}, y) true) - W (x_{1}, \dots, x_{d - 1}, y) false| = o_{p} false(1 false)

and

\underset{1 / n ⩽ u_{1}, u_{2} ⩽ 1 - 1 / n}{trueprefixsup} \frac{| \sqrt{n} (C_{n} false(u_{1}, u_{2}; j false) - C false(u_{1}, u_{2}; j false)) - W false(u_{1}, u_{2}; j false) |}{{false{\min (u_{1}, u_{2}, 1 - u_{1}, 1 - u_{2}) false}}^{δ}} = o_{p} false(1 false)

for any

δ \in (0, 1 / 2)

, where

W (u_{1}, u_{2}; j)

W (x_{1}, \dots, x_{d - 1}, y)

with

x_{j} = u

…”

Section: Nonparametric Inferencesmentioning

confidence: 99%

“…Then, it follows from proposition 4.4 of Berghaus et al. () that for any

δ \in (0, 1 / 2)

λ \in (0, 1)

and

δ_{n} \to 0

({, \begin{matrix} 0true sup_{0 < y < 1} \frac{| \sqrt{n} (U_{n} false(y false) - y) |}{y^{δ} {false(1 - y false)}^{δ}} = O_{p} (1), sup_{1 / n^{λ} < y ⩽ 1 - 1 / n^{λ}} \frac{| \sqrt{n} (U_{n}^{-} false(y false) - y) |}{y^{δ} {false(1 - y false)}^{δ}} = O_{p} (1) \\ 0true sup_{| u_{1} - u_{2} | + | v_{1} - v_{2} | ⩽ δ_{n}} \frac{| \sqrt{n} {C_{n} false(u_{1}, v_{1}; j false) - C false(u_{1}, v_{1}; j false)} - \sqrt{{} C_{n} (u_{2}, v_{2}; j) - C (u_{2}, v_{2}; j)} |}{\max false{false| u_{1} - u_{2} ...} \end{matrix})

…”

Section: Proofsunclassified

Section: Nonparametric Inferencesmentioning

confidence: 99%

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An efficient approach to quantile capital allocation and sensitivity analysis

Asimit

Peng

Wang

et al. 2019

Mathematical Finance

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In various fields of applications such as capital allocation, sensitivity analysis, and systemic risk evaluation, one often needs to compute or estimate the expectation of a random variable, given that another random variable is equal to its quantile at some prespecified probability level. A primary example of such an application is the Euler capital allocation formula for the quantile (often called the value‐at‐risk), which is of crucial importance in financial risk management. It is well known that classic nonparametric estimation for the above quantile allocation problem has a slower rate of convergence than the standard rate. In this paper, we propose an alternative approach to the quantile allocation problem via adjusting the probability level in connection with an expected shortfall. The asymptotic distribution of the proposed nonparametric estimator of the new capital allocation is derived for dependent data under the setup of a mixing sequence. In order to assess the performance of the proposed nonparametric estimator, AR‐GARCH models are proposed to fit each risk variable, and further, a bootstrap method based on residuals is employed to quantify the estimation uncertainty. A simulation study is conducted to examine the finite sample performance of the proposed inference. Finally, the proposed methodology of quantile capital allocation is illustrated for a financial data set.

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