Abstract
This article studies a new family of bivariate copulas constructed using the unit-Lomax distortion derived from a transformation of the non-negative Lomax random variable into a variable whose support is the unit interval. Existing copulas play the role of the base copulas that are distorted into new families of copulas with additional parameters, allowing more flexibility and better fit to data. We present general forms for the new bivariate copula function and its conditional and density distributions. The properties of the new family of the unit-Lomax induced copulas, including the tail behaviors, limiting cases in parameters, Kendall’s tau, and concordance order, are investigated for cases when the base copulas are Archimedean, such as the Clayton, Gumbel, and Frank copulas. An empirical application of the proposed copula model is presented. The unit-Lomax distorted copula models outperform the base copulas.
Highlights
Numerous data sets in the field of actuarial science, finance, and medicine contain random variables, such as stock indexes and returns, that cannot be treated under the assumption of independence
If the base copula is the independence copula defined as C (u, v) = uv with generator φ(u) = − log u, the unit Lomax (UL)-independence copula is expressed as CT (u, v)
We demonstrate by plotting Kendall’s tau values that the families of the UL-Clayton, UL-Gumbel, and UL-Frank copulas are not ordered by the parameters a and b stemming from the distortion function
Summary
Numerous data sets in the field of actuarial science, finance, and medicine contain random variables, such as stock indexes and returns, that cannot be treated under the assumption of independence. Sklar’s theorem (see Sklar (1959)) proves the existence of a unique copula that captures the dependence structures among continuous random variables This allows researchers to expand venues for modeling multivariate data in the real world; by using existing copulas, and by establishing new copulas (Nelsen (2006); Joe (2015)). If two continuous random variables, X and Y, with margins F and G have a joint distribution function (cdf) H,, there exists a unique copula C such that H ( x, y) = P( X ≤.
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