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Let $\hat{\operatorname{F}}_{n}$ be a sequence of estimators of a cumulative distribution $\operatorname{F}$:

  • If $\displaystyle\sup_{x \in \mathbb{R}} \left\vert\,{\hat{\operatorname{F}}_{n}\left(x\right) - \operatorname{F}\left(x\right)}\,\right\vert \to_{\mathbb{P}} 0,$
  • does it hold that for a fixed $\alpha \in \left(0,1\right),\ \hat{\operatorname{F}}^{-1}_{n}\left(\alpha\right) \to_{\mathbb{P}} \operatorname{F}^{-1}\left(\alpha\right)\, ?.$

You can assume all distributions are continuous.

Felix Marin
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madness
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1 Answers1

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In general, this only holds for values of $\alpha$ that are continuity points of the generalized inverse, which are those points at which $F$ is strictly increasing. If it is a continuity point, this follows from result that convergence in distribution implies convergence of the quantile function in continuity points (proof). For an example illustrating that the stronger mode of convergence does not improve the result see here.

user159517
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