I want to mathematically understand the effect of skewness and kurtosis in regards to $t$-statistic and $f$-statistic, for test of means and variances.
I read from here,
Under regularity assumptions, we obtain the following asymptotic expansion for the cdf of the test statistic $T_n$: $$P(T_n\leq x)=\Phi(x)+n^{-1/2}\frac{1}{6}\gamma(2x^2+1)\phi(x)-n^{-1}x\Big(\frac{1}{12}\kappa (x^2-3)-\frac{1}{18}\gamma^2(x^4+2x^2-3)-\frac{1}{4}(x^2+3)\Big)\phi(x)+o(n^{-1}),$$
where $\gamma$ is skewness, and $\kappa$ is kurtosis.
Is there any reference that explains this? Or can anyone verify if this is actually correct?
More importantly, if the $t$-statistic can be expanded like that, what would the expansion of $f$-statistic look like?
Could anyone derive the expansion of $f$-statistic, or give me a link that does it?