I would like to compute the following summation:

$$ s = \sum_{i=1}^n a_i \, \Phi^{-1}(u_i) $$

where $\Phi^{-1}$ is the inverse of the standard Gaussian distribution function, $a_i$ are some real numbers, and $u_i$ are in $[0, 1]$ so that $u_i = 0$ for some $i$s, $u_i = 1$ for some other $i$s, and $u_i \in (0, 1)$ for the rest. In other words, the sum has several finite elements, several elements equal to $+\infty$, and several elements equal to $-\infty$. As far as I understand, the result is undefined in general; however, we know what those infinities are.

My idea is to break the sum into three sums as follows:

$$ \begin{align*} s &= \sum_{i} a_i \, \Phi^{-1}(u_i) + \Phi^{-1}(1) \sum_{i} a_i + \Phi^{-1}(0) \sum_{i} a_i\\ &= s_1 + \Phi^{-1}(1) s_2 + \Phi^{-1}(0) s_3. \end{align*} $$

Then I am planning to proceed as follows. If $s_1 = s_2$, the sum is set to $s_1$. (This also includes the zero case, which is already arguable.) Depending on the signs and magnitudes of $s_1$ and $s_2$, the sum is set to $+\infty$ or $-\infty$.

My question is: Does the above procedure make sense? Is it legitimate to operate with infinities as I described?

I would appreciate any thoughts and suggestions. Thank you!

Regards, Ivan

  • $\begingroup$ When might you want to do this? $\endgroup$ – Henry Apr 28 '15 at 20:04
  • $\begingroup$ @Henry, I have a Gaussian copula and a vector in $[0, 1]^n$. I would like to investigate how the copula affects that particular vector. So, I apply $\Phi^{-1}$ to the vector and multiply it by a matrix obtained using the Cholesky decomposition or alike. $\endgroup$ – Ivan Apr 28 '15 at 20:15

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