Let us consider the statistic $$U_n = \frac{1}{\sqrt{n}} \sum_{i = 1}^n \frac{\epsilon_{i, n}}{\sigma_{i,n}} + f(\sigma_{1, n},\dots,\sigma_{n, n}),$$ where, conditional on $\sigma_{1, n},\dots,\sigma_{n, n}$, $\epsilon_{i, n} \overset{iid}{\sim}$ some distribution with zero mean and variance $\sigma^2_{i, n}$ and $f$ is a continuous function. The variables $\sigma_{1, n},\dots,\sigma_{n, n}$ are not independent.
I have an approximation of the asymptotic distribution of $f(\sigma_{1, n},\dots,\sigma_{n, n})$. I know the quantiles. But I am not sure that this is important for my question.
My purpose is to approximate the quantile of $U_n$ conditional on $\sigma_{1, n},\dots,\sigma_{n, n}$. This is how I proceed, but I am not sure whether it is theoretically correct. If not, how to adjust the result?
- Conditional on $\sigma_{1, n},\dots,\sigma_{n, n}$, I argued that $\frac{1}{\sqrt{n}} \sum_{i = 1}^n \frac{\epsilon_{i, n}}{\sigma_{i,n}}$ is asymptotically normally distributed. The intuition is that if the treat $\sigma_{1, n},\dots,\sigma_{n, n}$ as nonstochastic, the Lyapunov central limit theorem can be used if $$\frac{\sum_{i = 1}^n \mathbb{E}\lvert\epsilon_{i,n}\rvert^{2 + \delta}}{\sum_{i = 1}^n \sigma_{i,n}^2} = o_p(1).$$ Can the conditional asymptotic distribution be stated in this way?
- If the preceding point is correct, can I say that $\mathbb{P}(U_n < x|\sigma_{1, n},\dots,\sigma_{n, n})$ has the same limit as $\mathbb{P}(\zeta < x - f(\sigma_{1, n},\dots,\sigma_{n, n})|\sigma_{1, n},\dots,\sigma_{n, n})$, where $\zeta\sim N(0, 1)$?
What will be the implication of the result? I can finally estimate the unconditional distribution function as follows: $$\mathbb{E}(\mathbb{P}(\zeta < x - f(\sigma_{1, n},\dots,\sigma_{n, n})|\sigma_{1, n},\dots,\sigma_{n, n})).$$
I can estimate the above quantity by replacing the expectation by an empirical mean if I have realizations (simulations) of $\sigma_{1, n},\dots,\sigma_{n, n}$.