Consider $n$ i.i.d. (continues) random variables $X_1,...,X_n$ sampled a distribution with density function $f(x)$.
Let $g(x_1,...,x_n): \mathbb{R}^n\mapsto \mathbb{R}$ be a function defined over $x_1,...,x_n$. Define $G(n):=\int_{-\infty}^{\infty} g(x_1,...,x_n) \Pi_i^n f(x_i) dx_i$.
Consider the following optimization problem:
$$n^*:=\min \{n: G(n)>\xi\},$$
where $\xi$ is a real number between $(0,1)$.
This can be seen as to find the minimum sample size $n^*$ such that the statistic $T:=g(X_1,...,X_n)$ meets a certain probability constraint.
My question is how to solve the above optimization problem, when the function $g(\cdot)$ and the integral $G(n)$ are hard to be represented analytically?
As an example, consider $g(x_1,...,x_n):=\Phi_n(x_1,...,x_n)$ being the CDF of the $n$-variate Gaussian distribution (with diagonal covariance matrix) and $f(x)\sim N(0,1)$.
Any suggestion or reference is welcomed.