Bernstein estimator for unbounded copula densities

Bouezmarni, Taoufik; Ghouch, Anouar El; Taamouti, Abderrahim

doi:10.1524/strm.2013.2003

Cited by 21 publications

(18 citation statements)

References 27 publications

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“…It consists of minimizing with respect to m the estimated mean integrated squared error, estimated from the data using the leave-one out idea. For Bernstein copula density estimators, this method has been used, for example, by Bouezmarni et al (2013) and for Bernstein density estimators by Bouezmarni and Rolin (2007).…”

Section: Remark 1 the Bias Of Cmentioning

confidence: 99%

Bernstein estimation for a copula derivative with application to conditional distribution and regression functionals

Janssen

Swanepoel

Veraverbeke

2015

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Bernstein estimators attracted considerable attention as smooth nonparametric estimators for distribution functions, densities, copulas and copula densities. The present paper adds a parallel result for the first-order derivative of a copula function. This result then leads to Bernstein estimators for a conditional distribution function and its important functionals such as the regression and quantile functions. Results of independent interest have been derived such as an almost sure oscillation behavior of the empirical copula process and a Bahadur-type almost sure asymptotic representation for the Bernstein estimator of a regression quantile function. Simulations demonstrate the good performance of the proposed estimators.

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Section: Remark 1 the Bias Of Cmentioning

confidence: 99%

Bernstein estimation for a copula derivative with application to conditional distribution and regression functionals

Janssen

Swanepoel

Veraverbeke

2015

TEST

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“…Using Bernstein estimators for the survival copula and its derivatives, we obtain Bernstein based estimators for the conditional hazards and a nonparametric estimator for the cross ratio function θ(t 1 , t 2 ). The reason for using a Bernstein copula-based estimator for the cross ratio function is motivated from the results in the papers by Sancetta and Satchell (2004), Leblanc (2012), Janssen et al (2012Janssen et al ( , 2014Janssen et al ( , 2016 and Bouezmarni et al (2009Bouezmarni et al ( , 2013. Simulations in these papers show that, compared to its nonparametric competitors (including kernel estimators), Bernstein based estimators for the copula and copula derivatives are superior.…”

Section: Introductionmentioning

confidence: 99%

Nonparametric estimation of the cross ratio function

et al. 2019

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The cross ratio function (CRF) is a commonly used tool to describe local dependence between two correlated variables. Being a ratio of conditional hazards, the CRF can be rewritten in terms of (first and second derivatives of) the survival copula of these variables. Bernstein estimators for (the derivatives of) this survival copula are used to define a nonparametric estimator of the cross ratio and asymptotic normality thereof is established. We consider simulations to study the finite sample performance of our estimator for copulas with different types of local dependency. A real dataset is used to investigate the dependence between food expenditure and net income. The estimated CRF reveals that families with a low net income relative to the mean net income will spend less money to buy food compared to families with larger net incomes.

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“…The basic version of the Bernstein copula as developed by Sancetta and Satchell (2004) for iid data and extended by Bouezmarni et al (2010Bouezmarni et al ( , 2013 to dependent data and to unbounded densities is defined as follows:…”

Section: Bernstein Polynomial-based Estimatorsmentioning

confidence: 99%

“…Abbreviations: DME: data mirror estimator; EBCE: empirical beta copula density estimator; NKE: naive kernel estimator; PESE: penalized exponential estimator; SE: spline estimators; SMB: sieve maximum-likelihood estimation with Bernstein. In order to investigate the behaviour of the estimators near the boundary, we follow Bouezmarni et al (2013) and focus on upper right and lower left corners of the square. Specifically, we define two regions S 1 and S 2 of [0, 1] 2 as follows…”

Section: 4smentioning

confidence: 99%

A Simple Estimator of Two‐Dimensional Copulas, with Applications1

2020

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Copulas are distributions with uniform marginals. Non-parametric copula estimates may violate the uniformity condition in finite samples. We look at whether it is possible to obtain valid piecewise linear copula densities by triangulation. The copula property imposes strict constraints on design points, making an equi-spaced grid a natural starting point. However, the mixed-integer nature of the problem makes a pure triangulation approach impractical on fine grids. As an alternative, we study the ways of approximating copula densities with triangular functions which guarantees that the estimator is a valid copula density. The family of resulting estimators can be viewed as a non-parametric MLE of B-spline coefficients on possibly non-equally spaced grids under simple linear constraints. As such, it can be easily solved using standard convex optimization tools and allows for a degree of localization. A simulation study shows an attractive performance of the estimator in small samples and compares it with some of the leading alternatives. We demonstrate empirical relevance of our approach using three applications. In the first application, we investigate how the body mass index of children depends on that of parents. In the second application, we construct a bivariate copula underlying the Gibson paradox from macroeconomics. In the third application, we show the benefit of using our approach in testing the null of independence against the alternative of an arbitrary dependence pattern.

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Bernstein estimator for unbounded copula densities

Cited by 21 publications

References 27 publications

Bernstein estimation for a copula derivative with application to conditional distribution and regression functionals

Bernstein estimation for a copula derivative with application to conditional distribution and regression functionals

Nonparametric estimation of the cross ratio function

A Simple Estimator of Two‐Dimensional Copulas, with Applications1

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