Abstract
Copulas played a key role in numerous areas of statistics over the last few decades. In this paper, we offer a new kind of trigonometric bivariate copula based on power and cosine functions. We present it via analytical and graphical approaches. We show that it may be used to create a new bivariate normal distribution with interesting shapes. Subsequently, the simplest version of the suggested copula is highlighted. We discuss some of its relationships with the Farlie-Gumbel-Morgensten and simple polynomial-sine copulas, establish that it is a member of a well-known semi-parametric family of copulas, investigate its dependence domains, and show that it has no tail dependence.
Highlights
Modeling random multivariate events is a major topic in many scientific disciplines and creating multivariate distributions that can accurately model the variables at play is difficult
We focus on its properties, proving some ordering results involving the FGM and simple polynomial-sine (SPS) copulas, highlighting the link with a more general well-identified family of copulas, and investigating the related dependence domain and tail dependence
The corresponding simple PC (SPC) copula density is given as c (u, v) = 1 + λ π u sin π u − cos π u π v sin π v − cos π v and the medial correlation coefficient is reduced to M = λ/2
Summary
Modeling random multivariate events is a major topic in many scientific disciplines and creating multivariate distributions that can accurately model the variables at play is difficult. The major result on the concept of copula is the Sklar theorem established in [1]. It ensures that, if F(x, y) is a joint cumulative distribution function (CDF) with marginal CDFs given by FX(x) and FY(y), respectively, there exists a copula C(u, v) such that F(x, y) = C(FX(x), FY(y)). The lack of simple trigonometric copula in the literature is one of the motivations of this study, opening some new horizons for bivariate modelling. We use it to construct a new bivariate normal distribution with flexible bell shapes. The rest of the paper is conducted as follows: Section 2 defines the new copula, as well as its major features and direct applications.
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