How to prove that a function is 2-increasing (copula)

Asked Oct 27 '20 at 13:17

Active Oct 27 '20 at 15:36

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There are three conditions to prove that a function is a copula:

Here I am concerning in the last condition how to prove that a function is 2-increasing as example $H(x,y)= (2x-1)(2y-1)$.

Is it correct if the second derivative of $H(x,y)$ is greater than zero then the $H(x,y)$ is 2-increasing?

edited Oct 27 '20 at 15:36

develarist

asked Oct 27 '20 at 13:17

محمد عبد

what is $H(x,y)$? – develarist Oct 27 '20 at 15:36
It depends on which second derivative and on whether $H$ is differentiable everywhere on $[0,1]^2.$ – whuber Oct 27 '20 at 15:39
H(x,y) is a bivariate function defined on [0,1] x [0,1] – محمد عبد Oct 27 '20 at 16:41
It is a partial derivative ∂^2/∂x∂y @whuber – محمد عبد Oct 27 '20 at 16:45
Not all functions are differentiable. Indeed, many copulas are not differentiable. Just apply the definition of 2-increasing in any particular case. – whuber Oct 27 '20 at 17:03
suppose we have Farlie-Gumbel-Morgenstern copula the second derivative – محمد عبد Oct 27 '20 at 17:13
Dear @whubar, please tell me how to prove a differentiable function in a certain interval is 2-increasing? Moreover, if the second partial derivative of a function is non-negative, can we say the function is 2-increasing. If yes, please provide a reference. – محمد عبد Oct 27 '20 at 17:58

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