This question related to my previous question.

Let $$X_1,\dots,X_n$$ are i.i.d. with distribution function $F$ and $$Y_1,\dots,Y_n$$ are i.i.d. with distribution function $G$. Suppose that there exists an unknown increasing function $\psi:\mathbb{R}\mapsto\mathbb{R}$ such that $\psi(X_i)\sim N(0,1)$ and $\psi(Y_j)\sim N(0,\sigma^2)$ for all $i,j=1,\dots,n$. I wish to evaluate the variance of plug-in estimator of $\sigma^2$ in this problem.

I have obtained the following facts:

Note that $F(x)=P(X\le x)=P(\psi(X)\le \psi(x))=\Phi(\psi(x))$ where $\Phi$ is the cumulative standard normal distribution. This implies $\psi(x)=\Phi^{-1}(F(x))$.

Recall that $\psi(Y_j)\sim N(0,\sigma^2)$. So, $$\check\sigma^2=\frac1n\sum_{j=1}^n\psi^2(Y_j)$$ is an optimal estimator for $\sigma^2$. Since $F$ is unknown then I replace it with its empirical distribution function $\hat F_n$ based on $X_1,\dots,X_n$. Hence, it is natural to replace $\psi$ with $\hat\psi=\Phi^{-1}(\hat F_n)$. Therefore, I conjecture that $$\hat\sigma^2=\frac1n\sum_{j=1}^n\hat\psi^2(Y_j)$$ is the estimator that I wanted. The problem is I don't know what the variance of $\hat\sigma^2$ is. Any suggestion?



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