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Let $\{X_i\}_{i=1}^n$ be a random sample, $Y_i=\delta(X_i)$ for some function $\delta(\cdot)$, and $Q(\tau)$ be the population quantile function of $Y_i$. We can estimate $Q(\cdot)$ by the empirical quantile function $\widehat Q(\tau)$ based on the random sample $\{Y_i\}_{i=1}^n$. Under certain regularity conditions, we have \begin{align*} \sup_{\tau\in(0,1)}|\widehat Q(\tau)-Q(\tau)|=o_P(1). \end{align*}

Now suppose we do not know the function $\delta(\cdot)$. Instead, we have a uniform convergent estimator $\widetilde\delta(\cdot)$ for $\delta(\cdot)$, that is, \begin{align*} \sup_{x\in\mathcal{X}}|\widetilde\delta(x)-\delta(x)|=o_P(1). \end{align*}

Given the pseudo sample $\{\widetilde Y_i\}_{i=1}^n$, $\widetilde Y_i=\widetilde \delta(X_i)$, and let $\widetilde Q(\tau)$ be the empirical quantile function based on the pseudo sample, do we have \begin{align*} \sup_{\tau\in(0,1)}|\widetilde Q(\tau)-Q(\tau)|=o_P(1)? \end{align*} How to show this and what conditions are needed?

Thanks a lot for your help.

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