On copulas with a trapezoid support

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Abstract A family of bivariate copulas given by: for v + 2 u < 2 v+2u\lt 2 , C ( u , v ) = F ( 2 F − 1 ( v ∕ 2 ) + F − 1 ( u ) ) C\left(u,v)=F\left(2{F}^{-1}\left(v/2)+{F}^{-1}\left(u)) , where F F is a strictly increasing cumulative distribution function of a symmetric, continuous random variable, and for v + 2 u ≥ 2 v+2u\ge 2 , C ( u , v ) = u + v − 1 C\left(u,v)=u+v-1 , is introduced. The basic properties and necessary conditions for absolute continuity of C C are discussed. Several examples are provided.