Moment expansion of likelihood function

Question

Let's say I have a binary hypothesis test described by a probability distribution $p(x;\theta)$ where $\theta$ takes two different values under each of the two hypothesis ($\theta_0$ under $\mathcal{H}_0$, the null hypothesis, and $\theta_1$ under $\mathcal{H}_1$). Using the Neyman-Pearson theorem, the test decide $\mathcal{H}_1$ if the likelihood is greater than a given threshold $\gamma$ $$\mathcal{L}(x)=\frac{p(x;\theta_1)}{p(x;\theta_0)}>\gamma$$ But in my situation, I know only the moments $m_n=\left<x^n\right>$ or, equivalently, the cumulants $k_n=\left<\left<x^n\right>\right>$ of the distribution. Is there a way to expand the likelihood functions in series of its moments?

Answer 1

I think I've found the solution. It is called Gram-Charlier expansion. link