# Is $H=\min(t_1,…,t_n)$ a Copula?

n-Box is defined as $B=[a_1,b_1]\times[a_2,b_2]\times[a_3,b_3]\times...\times[a_n,b_n]$ Cartesian product of $n$ closed intervals, where $a_i$ and $b_i$ are all from $R$ and $a_i\le b_i$ for $i=1,...,n$. Vertices of the box are $c=(c_1,...,c_n)$ where $c_k=a_k$ or $c_k=b_k$.

We have a function defined as $H=\min(t_1,...,t_n)$.

We define $H$ volume of the box as $V_H(B)=\sum \mathrm{sgn}(c)H(c)$ where the sum is taken over all vertices of the box.

$\mathrm{sgn}(c)=1$ if $c_k=a_k$ for an even number of $k$'s

$\mathrm{sgn}(c)=-1$ if $c_k=a_k$ for an odd number of $k$'s

Prove that $V_H(B)=\max(\min(b_1,...,b_n)-\max(a_1,...,a_n),0)$

That would also be a proof that $H$ is Copula.

• Then you should probably add the self-study tag, and read its tag wiki info. – Glen_b Jun 22 '14 at 12:38