A note on nonparametric estimation of copula-based multivariate extensions of Spearman’s rho

Pérez, Ana; Alaiz, Mercedes Prieto

doi:10.1016/j.spl.2016.01.015

Cited by 15 publications

(16 citation statements)

References 13 publications

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“…In particular, these authors propose estimating nonparametrically the coefficients

ρ_{d}^{-}

and

ρ_{d}^{+}

by replacing the copula C in (3) and (4), respectively, with the empirical copula in (10). However, Pérez and Prieto‐Alaiz (2016b) show that the resultant statistics are not proper estimators of their population counterparts, since they can take values out of the parameter space. The modifications proposed by Blumentritt and Schmid (2014) and Bedo and Ong (2014) have still some drawbacks, as they fail to achieve the maximum value 1 for maximal dependence and take a narrower range of values than they should.…”

Section: Methodsmentioning

confidence: 99%

“…The modifications proposed by Blumentritt and Schmid (2014) and Bedo and Ong (2014) have still some drawbacks, as they fail to achieve the maximum value 1 for maximal dependence and take a narrower range of values than they should. To overcome these problems, Pérez and Prieto‐Alaiz (2016b) propose alternative feasible nonparametric estimators of

ρ_{d}^{-}

and

ρ_{d}^{+}

, based on the results in Joe (1990), which are given by the following expressions, respectively:

{\overset{ρ}{true}}_{d}^{-} = \frac{\frac{1}{n} {false\sum}_{j = 1}^{n} {false\prod}_{i = 1}^{d} {\overset{\bar{U}}{true}}_{italicij} - {()(, \frac{n + 1}{2 n})}^{d}}{\frac{1}{n} {false\sum}_{j = 1}^{n} {()(, \frac{j}{n})}^{d} - {()(, \frac{n + 1}{2 n})}^{d}},

{\overset{ρ}{true}}_{d}^{+} = \frac{\frac{1}{n} {false\sum}_{j = 1}^{n} {false\prod}_{i = 1}^{d} {\overset{U}{true}}_{italicij} - {()(, \frac{n + 1}{2 n})}^{d}}{\frac{1}{n} {false\sum}_{j = 1}^{n} {()(, \frac{j}{n})}^{d} - {()(, \frac{n + 1}{2 n})}^{d}},

where

{\overset{\bar{U}}{true}}_{italicij} = {\bar{R}}_{italicij} / n

and

R

…”

Section: Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Copula‐based analysis of multivariate dependence patterns between dimensions of poverty in Europe*

García‐Gómez

Pérez

Prieto‐Alaiz

2020

Review of Income and Wealth

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It is widely recognized that poverty is a multidimensional phenomenon involving not only income, but also other aspects such as education or health. In this multidimensional setting, analyzing the dependence between dimensions becomes an important issue, since a high degree of dependence could exacerbate poverty. In this paper, we propose measuring the multivariate dependence between the dimensions of poverty in Europe using copula‐based methods. This approach focuses on the positions of individuals across dimensions, allowing for other types of dependence beyond linear correlation. In particular, we analyze how orthant dependence between the dimensions of the AROPE rate has evolved in the EU‐28 countries between 2008 and 2014 by applying non‐parametric estimates of multivariate copula‐based generalisations of Spearman’s rank correlation coefficient. We find a general increase in the dependence between dimensions, regardless of the coefficient used. Moreover, countries with higher AROPE rates also tend to experiment more dependence between its dimensions.

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“…In particular, these authors propose estimating nonparametrically the coefficients

ρ_{d}^{-}

and

ρ_{d}^{+}

Section: Methodsmentioning

confidence: 99%

ρ_{d}^{-}

and

ρ_{d}^{+}

, based on the results in Joe (1990), which are given by the following expressions, respectively:

{\overset{ρ}{true}}_{d}^{-} = \frac{\frac{1}{n} {false\sum}_{j = 1}^{n} {false\prod}_{i = 1}^{d} {\overset{\bar{U}}{true}}_{italicij} - {()(, \frac{n + 1}{2 n})}^{d}}{\frac{1}{n} {false\sum}_{j = 1}^{n} {()(, \frac{j}{n})}^{d} - {()(, \frac{n + 1}{2 n})}^{d}},

{\overset{ρ}{true}}_{d}^{+} = \frac{\frac{1}{n} {false\sum}_{j = 1}^{n} {false\prod}_{i = 1}^{d} {\overset{U}{true}}_{italicij} - {()(, \frac{n + 1}{2 n})}^{d}}{\frac{1}{n} {false\sum}_{j = 1}^{n} {()(, \frac{j}{n})}^{d} - {()(, \frac{n + 1}{2 n})}^{d}},

where

{\overset{\bar{U}}{true}}_{italicij} = {\bar{R}}_{italicij} / n

and

R

…”

Section: Methodsmentioning

confidence: 99%

Copula‐based analysis of multivariate dependence patterns between dimensions of poverty in Europe*

García‐Gómez

Pérez

Prieto‐Alaiz

2020

Review of Income and Wealth

Self Cite

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“…where the lower bound was introduced by [18], and was shown to be the best possible lower bound by Úbeda-Flores [33,Theorem 5.1]. Note that the lower bound in (19) converges to zero as d tends to infinity. The minimal value of τ is achieved when [0,1] d C d (u) dC d (u) is equal to zero which happens for example for a random vector containing a random variable X and also −X.…”

Section: Multivariate Kendall's Taumentioning

confidence: 99%

On the specification of multivariate association measures and their behaviour with increasing dimension

Gijbels

Kika

Omelka

2021

Journal of Multivariate Analysis

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“…Subsequently, one can calculate

The estimator of

that could then be considered is of the form

However, this quantity does not provide the value 1 for a sample from a comonotonicity copula. See the related discussion in [ 25 ]. This problem increases, while

gets smaller.…”

Section: Estimation Of Tail Coefficientsmentioning

confidence: 99%

Multivariate Tail Coefficients: Properties and Estimation

Gijbels

Kika

Omelka

2020

Entropy

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Multivariate tail coefficients are an important tool when investigating dependencies between extreme events for different components of a random vector. Although bivariate tail coefficients are well-studied, this is, to a lesser extent, the case for multivariate tail coefficients. This paper contributes to this research area by (i) providing a thorough study of properties of existing multivariate tail coefficients in the light of a set of desirable properties; (ii) proposing some new multivariate tail measurements; (iii) dealing with estimation of the discussed coefficients and establishing asymptotic consistency; and, (iv) studying the behavior of tail measurements with increasing dimension of the random vector. A set of illustrative examples is given, and practical use of the tail measurements is demonstrated in a data analysis with a focus on dependencies between stocks that are part of the EURO STOXX 50 market index.

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A note on nonparametric estimation of copula-based multivariate extensions of Spearman’s rho

Abstract: Schmid and Schmidt (2007) proposed copula-based nonparametric estimators for some multivariate extensions of Spearman's rho. In this paper, we show that two of those estimators are inappropriate since they can take values out of the parameter space and we discuss alternative proposals.

Cited by 15 publications

References 13 publications

Copula‐based analysis of multivariate dependence patterns between dimensions of poverty in Europe*

Copula‐based analysis of multivariate dependence patterns between dimensions of poverty in Europe*

On the specification of multivariate association measures and their behaviour with increasing dimension

Multivariate Tail Coefficients: Properties and Estimation

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