I have a univariate cumulative distribution function (CDF) that has the form
$$\Pr (X \leq x) = F_X(x) = \int_{-\infty}^xh(s) e^{-s^2}ds$$
Namely it very much looks like having the appropriate form for Gauss-Hermite quadrature, except the range of integration. "$x$" will be a fixed known number.
I have searched in vain to find any source on the matter.
When people write about Gauss-Hermite quadrature, they always consider the $(-\infty, \infty)$ interval.
I guess I could do something like
$$\Pr (X \leq x) = 1- P(X> x) = 1 - \int_x^{\infty}h(s) e^{-s^2}ds$$
$$ = 1 - \int_x^{0}h(s) e^{-s^2}ds - \int_0^{\infty}e^{-s}\cdot [h(s) e^{-s^2}e^{s}]ds$$
and consider a transformation of the first integral to apply Gauss-Legendre, and apply Gauss-Laguerre on the second, but this appears a totally twisted approach... Just a note, $x$ may be higher than zero in which case the first integral will be on $[0,x]$ and will change sign in front.
Can I somehow apply Gauss-Hermite on the original expression for the CDF? Any sources on that? Any other suggestions?
I think I 've figured something out, that is a bit less twisted, but does not use Gauss-Hermite.
$$\Pr (X \leq x) = 1- P(X> x) = 1 - \int_x^{\infty}h(s) \exp\{-s^2\}ds$$
$$=1-\int_0^{\infty}e^{-s}\cdot [e^{s}h(s+x) \exp\{-(s+x)^2\}]ds$$
$$\implies \Pr (X \leq x) \approx 1- \sum_{j=1}^m w_jh(t_j+x) \exp\{-(t_j+x)^2+t_j\} $$
...where $w_j$ are the weights and $t_j$ the nodes for Gauss-Laguerre quadrature.