Editing Quantile (section)

==The asymptotic distribution of the sample median==
The sample median is the most examined one amongst quantiles, being an alternative to estimate a [[location parameter]], when the expected value of the distribution does not exist, and hence the sample mean is not a meaningful estimator of a population characteristic. Moreover, the sample median is a more robust estimator than the sample mean.

One peculiarity of the sample median is its asymptotic distribution: when the sample comes from a continuous distribution, then the sample median has the anticipated Normal asymptotic distribution,

: <math>\text{Sample median m} \sim \mathcal{N}\left(\mu=m, \sigma^2=\frac{1}{4Nf(m)^2}\right)</math>

This extends to the other quantiles,

: <math>\text{Sample quantile p} \sim \mathcal{N}\left(\mu=x_p, \sigma^2=\frac{p( 1 - p )}{Nf(x_p)^2}\right)</math>

where {{math|''f''(''x<sub>p</sub>'')}} is the value of the distribution density at the {{mvar|p}}-th population quantile (<math>x_p=F^{-1}(p)</math>).<ref name="Stuart1994">{{cite book |title=Kendall's Advanced Theory of Statistics |first1=Alan |last1=Stuart |first2=Keith |last2=Ord |location=London |publisher=Arnold |year=1994 |isbn=0340614307 }}</ref>  

But when the distribution is discrete, then the distribution of the sample median and the other quantiles fails to be Normal (see examples in https://stats.stackexchange.com/a/86638/28746).

A solution to this problem is to use an alternative definition of sample quantiles through the concept of the "mid-distribution" function, which is defined as 

: <math>F_\text{mid}(x) = P(X\le x) - \frac 12P(X=x)</math>

The definition of sample quantiles through the concept of mid-distribution function can be seen as a generalization that can cover as special cases the continuous distributions. For discrete distributions the sample median as defined through this concept has an asymptotically Normal distribution, see Ma, Y., Genton, M. G., & Parzen, E. (2011). Asymptotic properties of sample quantiles of discrete distributions. Annals of the Institute of Statistical Mathematics, 63(2), 227–243.