$G(x)$ is a Normal distribution with mean $\mu$ and standard deviation $\sigma$.

I observe realization of $X$ which are a function of $s$. The distribution $F(s)$ is found as the root (between 0 and 1) of $G_{n:n}(x) - n t^{n-1} - (n-1) t ^n = 0$, where $n$ is the number of observations and where $G_{n:n}(x)$ is the distribution of the highest order statistic of $X$ (whose parameters are known).

Info: this formula is just the numerical counterpart of the inverse of the second-order statistic formula. That is, I know that $G_{n:n}(x(s)) = F_{n-1:n}(s)$, so $F(s) = \phi^{-1}(G_{n:n})$. That formula is $\phi^{-1}$.

My question is: given that I know the parameters of the distribution $G(\cdot)$ how can I find the parameters of the distribution $F(s)$? Also, $x(.)$ is a function of $s$, and $G(x)$ is normal. Is $F(s)$ still normal?


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