Probit transformation for nonparametric kernel estimation of the copula density

Geenens, Gery; Charpentier, Arthur; Paindaveine, Davy

doi:10.48550/arxiv.1404.4414

Cited by 4 publications

(14 citation statements)

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“…in the proof of Theorem 1 and Proposition 3 we furthermore know that c je,ke;De (u, v) = c je,ke;De (u, v)+ o p {b 2 n,c + (nb 2 n,c ) −1/2 } as well as F je|De (x je |x De ) = F je|De (x je |x De ) + o p {b 2 n,c + (nb 2 n,c ) −1/2 }. Similar to Lemma B3, we can now show that c je,ke;De F je|De (x je |x De ), F ke|De (x ke |x De ) = c je,ke;De F je|De (x je |x De ), F ke|De (x ke |x De ) + o p {b 2 n,c + (nb 2 n,c ) −1/2 }.Hence, for(16) to hold it suffices to show that(nb 2 n,c ) 1/2 c * (x) − b 2 n,c μx − c * (x) d → N 0, Σx ,(25)wherec * (x) = cje,ke;De {F je|De (x je |x De ), F ke|De (x ke |x De )} e∈E 1 ,...,E d−1 , and c * (x) is defined similarly, but replacing c je,ke;De with c je,ke;De . ke|De ), z je|De := Φ −1 F je|De (x je |x De ) , z ke|De := Φ −1 F ke|De (x ke |x De ) .…”

supporting

confidence: 59%

“…C2 is less standard as it relates to the transformed density. Sufficient conditions for C2 are given in Lemma A.1 of [16] and can be verified for many parametric families, including the ones used in our simulation study.…”

Section: Estimation Of Pair-copula Densitiesmentioning

confidence: 99%

“…An estimator that takes no account of this property will suffer from bias issues at the boundaries of the support. A few kernel estimators particularly suited for bivariate copula densities were proposed in the literature [19,11,16]. Other nonparametric estimators can be constructed based on Bernstein polynomials [43], B-splines [31], or wavelets [17].…”

Section: Estimation Of Pair-copula Densitiesmentioning

confidence: 99%

“…This implies (nb 2 n,c ) 1/2 f (x) − f (x) = o p (1) and we have established that the first d components of ( 16) converge to zero in probability. Hence, the first d components of µ x as well as the first d rows and columns of Σ x will be zero and we only have to deal with the remaining components in (16).…”

Section: Proof Of Propositionmentioning

confidence: 99%

See 3 more Smart Citations

Evading the curse of dimensionality in nonparametric density estimation with simplified vine copulas

Nagler

Czado

2016

Journal of Multivariate Analysis

132

118

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Practical applications of nonparametric density estimators in more than three dimensions suffer a great deal from the well-known curse of dimensionality: convergence slows down as dimension increases. We show that one can evade the curse of dimensionality by assuming a simplified vine copula model for the dependence between variables. We formulate a general nonparametric estimator for such a model and show under high-level assumptions that the speed of convergence is independent of dimension. We further discuss a particular implementation for which we validate the high-level assumptions and establish its asymptotic normality. Simulation experiments illustrate a large gain in finite sample performance when the simplifying assumption is at least approximately true. But even when it is severely violated, the vine copula based approach proves advantageous as soon as more than a few variables are involved. Lastly, we give an application of the estimator to a classification problem from astrophysics.

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supporting

confidence: 59%

Section: Estimation Of Pair-copula Densitiesmentioning

confidence: 99%

Section: Estimation Of Pair-copula Densitiesmentioning

confidence: 99%

Section: Proof Of Propositionmentioning

confidence: 99%

See 2 more Smart Citations

Evading the curse of dimensionality in nonparametric density estimation with simplified vine copulas

Nagler

Czado

2016

Journal of Multivariate Analysis

132

118

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“…To estimate the density f , Nagler and Czado (2016) proposed to use the classical bivariate kernel density estimator (see, e.g., Silverman, 1986). We will extend this approach by resorting to the more general class of local polynomial likelihood estimators; see Loader (1999) for a general account and Geenens et al (2014) in the context of bivariate copula estimation.…”

Section: Kernel Weighted Local Likelihoodmentioning

confidence: 99%

Nonparametric estimation of simplified vine copula models: comparison of methods

Nagler

Schellhase

Czado

2017

Dependence Modeling

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In the last decade, simplified vine copula models have been an active area of research. They build a high dimensional probability density from the product of marginals densities and bivariate copula densities. Besides parametric models, several approaches to nonparametric estimation of vine copulas have been proposed. In this article, we extend these approaches and compare them in an extensive simulation study and a real data application. We identify several factors driving the relative performance of the estimators. The most important one is the strength of dependence. No method was found to be uniformly better than all others. Overall, the kernel estimators performed best, but do worse than penalized B-spline estimators when there is weak dependence and no tail dependence.

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Semiparametric copula quantile regression for complete or censored data

Backer¹,

Ghouch²,

Keilegom³

2017

Electron. J. Statist.

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When facing multivariate covariates, general semiparametric regression techniques come at hand to propose flexible models that are unexposed to the curse of dimensionality. In this work a semiparametric copula-based estimator for conditional quantiles is investigated for complete or right-censored data. In spirit, the methodology is extending the recent work of Noh et al. (2013) andNoh et al. (2015), as the main idea consists in appropriately defining the quantile regression in terms of a multivariate copula and marginal distributions. Prior estimation of the latter and simple plug-in lead to an easily implementable estimator expressed, for both contexts with or without censoring, as a weighted quantile of the observed response variable. In addition, and contrary to the initial suggestion in the literature, a semiparametric estimation scheme for the multivariate copula density is studied, motivated by the possible shortcomings of a purely parametric approach and driven by the regression context. The resulting quantile regression estimator has the valuable property of being automatically monotonic across quantile levels, and asymptotic normality for both complete and censored data is obtained under classical regularity conditions. Finally, numerical examples as well as a real data application are used to illustrate the validity and finite sample performance of the proposed procedure.

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Probit transformation for nonparametric kernel estimation of the copula density

Cited by 4 publications

References 0 publications

Evading the curse of dimensionality in nonparametric density estimation with simplified vine copulas

Evading the curse of dimensionality in nonparametric density estimation with simplified vine copulas

Nonparametric estimation of simplified vine copula models: comparison of methods

Semiparametric copula quantile regression for complete or censored data

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