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When studying and proving properties of sample quantiles, such as its consistency or its asymptotic normality, every text I have seen uses the standard definition of this estimators. This is, given a random sample $X_1, \ldots, X_n$ from a population, the sample quantile of order $p$ is defined as the quantile of this order of the empirical distribution, taking a convex combination (usually the midpoint) of two adjacent order statistics whenever the previous quantile is not uniquely defined.

However, different definitions are usually used in statistical packages, as they have better properties (a list and study of this definitions can be found here: https://www.amherst.edu/media/view/129116/original/Sample+Quantiles.pdf).

I would like to know whether the properties of the first sample quantiles are also true when using the other definitions.

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  • $\begingroup$ Which particular properties do you refer at? $\endgroup$ – usεr11852 Feb 1 '16 at 7:24

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