2
$\begingroup$

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?


$\endgroup$
0

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.