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I am interested in finding a closed form solution (wich I suspect does not exist) to the following integral

$$\displaystyle \int _a^{\infty }\int _b^{\infty } \frac{\exp \left(-\frac{x^2+y^2-2 c x y}{2 \left(1-c^2\right)}\right)}{2 \pi \sqrt{1-c^2}} dy dx$$

which corresponds to the integral of the PDF$(x,y)$ of a multiNormalDistribution (of covariance coefficient $c$) over the quarter plane $x>a$ and $y>b$. Here $a$ and $b$ are positive and $0 (and I know a solution exists for $a=b=0$, but this is not sufficient for my purpose).

More generally I would be interested in the $3$D generalization of this problem.

I have tried in Mathematica to no avail.

Regards,

1 Answers 1

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Generically, cumulative distribution function of multivariate Gaussian vector is not expressible in terms of cdf of standard normal random variable $\Phi(x)$. The book by Alan Genz and Frank Brentz, "Computation of multivariate Normal and t Probabilities" is good reference on the subject.

For a standard 2D Gaussian vector $(X,Y)$ with correlation coefficient $-1 < \rho <1$, the probability $ \mathbb{P}(X>a,Y>b) $ can be expressed in terms of Owen's T-function.

By the way Mathematica v8 has a built-in support for multi-normal distribution with special efficient cases for 2D and 3D Gussian random vectors, see BinormalDistribution (ref-page), and MultinormalDistribution (ref-page), and OwenT (ref-page).

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    Is it correct to assume there is no generalization of the Owen's T function which would be applicable to the 3D case?2012-06-04
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    @chris Of course there is. The generalization is called the multinormal probability. Generically it does not reduce to lower-dimensional probability functions though.2012-06-04
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    Nor does its cumulative distribution reduces to any analytic form?2012-06-04
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    No, I am afraid it does not2012-06-04
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    @Sasha Hey Sasha, a quick question. Is there a closed form expression for the CDF of bivariate gaussian in terms of Owen'sT-function? Or can the integral presented by OP be written in closed form in terms of Owen's T-function?2017-07-04
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    @AdelBibi Yes, because $F_{X,Y}\left(x,y)\right) = \Pr\left( X \leqslant x, Y \leqslant y\right) = 1 - \Pr\left(X > x\right) - \Pr\left(Y>y\right) + \Pr\left(X>x, Y>y\right) = F_X\left(x\right) +F_Y\left(y\right) - \left(1- \Pr\left(X>x, Y>y\right)\right)$. The latter bivariate probability can be expressed in terms of Owen's T-function as I claimed in the post.2017-07-04
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    @Sasha Thanks, Sasha. Do you by any means have any source of the last joint CDF in terms of Owen's T-function? I tried looking for it, but it is poorly documented.2017-07-05
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    @AdelBibi Please look at `OwenT` [ref-page](https://reference.wolfram.com/language/ref/OwenT.html), last example in the application section.2017-07-05