Multivariate Rényi inaccuracy measures based on copulas: properties and application

TL;DR

This work advances multivariate information measures by introducing Rényi inaccuracy measures based on copulas, namely MCCRI and MSCRI, to quantify how a proposed multivariate model deviates from a reference in terms dependence structure. It extends these ideas to co-copula and dual copula frameworks (MCoCRI and MDCRI) and derives bounds and comparison results under lower/upper orthant orders, including symmetry considerations. A semiparametric estimator for MCCRI is developed and validated via simulations, demonstrating reliable performance with increasing sample size. The authors also illustrate the practical utility of MCCRI as a model-selection criterion on real data, showing its potential to guide copula-based dependence modelling in multivariate settings.

Abstract

We propose Rényi inaccuracy measure based on multivariate copula and multivariate survival copula, respectively dubbed as multivariate cumulative copula Rényi inaccuracy measure and multivariate survival copula Rényi inaccuracy measure. Bounds of multivariate cumulative copula Rényi inaccuracy and multivariate survival copula Rényi inaccuracy measures have been obtained using Fréchet-Hoeffding bound. We discuss the comparison studies of the multivariate cumulative copula Rényi inaccuracy and multivariate survival copula Rényi inaccuracy measures based on lower orthant and upper orthant orders. We have also proposed multivariate co-copula Rényi inaccuracy and multivariate dual copula Rényi inaccuracy measures based on multivariate co-copula and dual copula. Similar properties have been explored. Further, we propose semiparametric estimator of multivariate cumulative copula Rényi inaccuracy measure. A simulation study is performed to compute standard deviation, absolute bias and mean squared error of the proposed estimator. Finally, a data set is considered to show that the multivariate cumulative copula Rényi inaccuracy measure can be applied as a model (copula) selection criteria.

Figures (3)

• Figure 1: Plots of the MCCRI and CCI measures of FGM and AHM copulas $(a)$ with respect to $\theta$ for $\alpha=0.5,~\lambda_1=0.8,~\lambda_2=1.5$ and $\gamma=1.5$; $(b)$ with respect to $\alpha$ for $\theta=0.5,~\lambda_1=0.8,~\lambda_2=1.5$ and $\gamma=1.5;$$(c)$ with respect to $\lambda_1$ for $\theta=0.8,~\alpha=0.5,~\lambda_2=4$ and $\gamma=3$, $(d)$ with respect to $\lambda_2$ for $\theta=0.5,~\alpha=0.8,~\lambda_1=1.5$ and $\gamma=1.5$ in Example \ref{['ex3.1']}.
• Figure 2: Plots of the MSCRI and SCI measures of FGM and AHM copulas $(a)$ with respect to $\theta$ for $\alpha=0.5,~\lambda_1=2,~\lambda_2=0.9$ and $\gamma=4$, $(b)$ with respect to $\alpha$ for $\theta=0.5,~\lambda_1=0.8,~\lambda_2=1.5$ and $\gamma=1.5$, $(c)$ with respect to $\lambda_1$ for $\theta=0.8,~\alpha=0.5,~\lambda_2=4$ and $\gamma=3$, $(d)$ with respect to $\lambda_2$ for $\theta=0.7,~\alpha=0.9,~\lambda_1=3$ and $\gamma=4$, in Example \ref{['ex4.1']}.
• Figure 3: Plots of the CoCCRI and DCRI measures of Joe and AHM copulas $(a)$ with respect to $\theta$ for $\alpha=0.9,~\lambda_1=3,~\lambda_2=0.7$ and $\gamma=4$, $(b)$ with respect to $\alpha$ for $\theta=2,~\lambda_1=3,~\lambda_2=0.7$ and $\gamma=4,$$(c)$ with respect to $\lambda_1$ for $\theta=2,~\alpha=0.6,~\lambda_2=0.7$ and $\gamma=4$, $(d)$ with respect to $\lambda_2$ for $\theta=3,~\alpha=0.9,~\lambda_1=1.7$ and $\gamma=4$ in Example \ref{['ex5.1']}.

Theorems & Definitions (44)

• Definition 2.1
• Theorem 2.1
• Definition 2.2
• Definition 2.3
• Definition 3.1
• Example 3.1
• Proposition 3.1
• proof
• Proposition 3.2
• proof