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In the standard quantile regression (QR) framework, we typically consider only one quantile level of interest, say $\tau$. By standard asymptotic results, we obtain the asymptotic normality of $\sqrt{n}(\hat{\beta}(\tau) - \beta(\tau))$, where $\hat{\beta}(\tau)$ is the standard QR estimator from Koenker and Bassett (1978). This gives us a way to perform inference for large samples using an estimate of the asymptotic variance.

However, when reading papers I sometimes also encounter asymptotic results that hold uniformly over all quantile levels $\tau \in (0, 1)$. These results do not seem to be a trivial extension of the one quantile level results. For example, in the Journal of Econometrics paper by Belloni et al. (2019) it is mentioned that

We note that inference on the function $u \to \beta(u)$, in particular simultaneous inference on the parameters $\beta(u)$ that holds uniformly over all $u ∈ U$, where $U$ is a set of quantile indices of interest, is difficult because the standard asymptotic theory (van de Vaart and Wellner, 1996) based on limit distributions does not help here as the process $u \to \sqrt{n}(\hat{\beta}(u)−\beta(u))$ in general does not have a limit distribution, even after an appropriate normalization.

This is only one of multiple papers I have seen that develop such uniform results over all quantile indices. I have taken a look at Van der Vaart and Wellner (1996) but I did not exactly find what is the reason that $u \to \sqrt{n}(\hat{\beta}(u)−\beta(u))$ does not have a limit. Does anyone have any ideas why this is the case? I want to build some intuition why the standard techniques do not work and how to derive results that hold uniformly. Any help or references to other works that explain this in more detail would be greatly appreciated.

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I am not an expert in quantile regression, but here is a general comment about convergence of stochastic processes.

In classical function spaces, a sequence $(S_n)$ of stochastic processes converges in distribution to a limit $S$, if and only if the finite-dimensional marginals (fidis) of $S_n$ converge to the fidis of $S$ and morover $(S_n)$ is a tight sequence in the function space. This is essentially a direct consequence of Prokhorov's theorem and the fact that fidis determine the joint distribution (the fidis form a separating class).

For you question, I think that tightness might be problematic. To build intuition, we can consider compactly supported and continuous distributions such that the quantile functions live in C[0,1], which we endow with the uniform norm. In this space, the tightness condition for a sequence $S_n(\cdot)$ is equivalent to

  1. There exists $a>0$ such that for any $\epsilon>0$ it holds that $P(|S_n(0)|\geq a)\leq \epsilon$ for $n$ sufficiently large.
  2. For any $\epsilon>0$: $\lim_{\delta\to 0^+}\limsup_{n\to \infty}P(\sup_{|t-s|\leq \delta}|S_n(t)-S_n(s)|\geq \epsilon)=0$.

In your case, I believe these correspond to some restrictive conditions on the distribution of interest.

  1. will likely require some sort lower bound on the support of the distribution,
  2. will likely require some type of strict positivity assumption, i.e., that the density is lower bounded by a constant.
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  • $\begingroup$ Thank you Alexander! I indeed remember a similar result from Van der Vaart & Wellner (Theorem 1.5.4) where they have such a result for non-measurable maps (that map to some function space). Probably it has something to do with the tightness condition then. Hopefully someone else has some further ideas how this applies to quantile regression. $\endgroup$
    – Stan
    Apr 11 at 12:06

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