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Please help me prove the following:

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.

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  • $\begingroup$ Is this for some subject/course/study/self-study? An exercise from a book, etc? $\endgroup$ – Glen_b Jun 22 '14 at 2:13
  • $\begingroup$ It is an exercise from Roger Nelsen "An introduction to Copulas" 2nd edition (Exercise 2.35). $\endgroup$ – Sergey Zykov Jun 22 '14 at 11:23
  • $\begingroup$ Then you should probably add the self-study tag, and read its tag wiki info. $\endgroup$ – Glen_b Jun 22 '14 at 12:38
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One way to prove a function $H$ on the $n$ cube $[0,1]^n$ is a copula is to exhibit a random variable whose distribution function restricted to the cube is $H.$

To that end, let $X$ be a univariate random variable with a uniform distribution on $[0,1],$ which means that for all $t\in[0,1],$ $\Pr(X\le t)=t.$ Define the $n$-vector-valued variable $\mathbf X$ as

$$\mathbf{X} = (X,X,\ldots, X).$$

Let $(t_1,\ldots, t_n)\in [0,1]^n$ and note (to justify the third equality below) that $\min(t_1,\ldots, t_n)\in[0,1].$ Successive application of the definitions of $\mathbf X,$ $\min,$ the uniform distribution, and $H$ justifies these four equalities:

$$\begin{aligned} \Pr(X_1\le t_1, \ldots, X_n\le t_n) &= \Pr(X \le t_1, \ldots, X\le t_n) \\ &= \Pr(X \le \min(t_1,\ldots,t_n)) \\ &= \min(t_1,\ldots, t_n) \\ &= H(t_1,\ldots, t_n), \end{aligned}$$

QED.

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