Editing Order statistic (section)

=== Large sample sizes ===

For the uniform distribution, as ''n'' tends to infinity, the ''p''<sup>th</sup> sample quantile is asymptotically [[normal distribution|normally distributed]], since it is approximated by

: <math>U_{(\lceil np \rceil)} \sim AN\left(p,\frac{p(1-p)}{n}\right).</math>

For a general distribution ''F'' with a continuous non-zero density at ''F''<sup>&nbsp;−1</sup>(''p''), a similar asymptotic normality applies:

: <math>X_{(\lceil np \rceil)} \sim AN\left(F^{-1}(p),\frac{p(1-p)}{n[f(F^{-1}(p))]^2}\right)</math>

where ''f'' is the [[density function]], and ''F''<sup>&nbsp;−1</sup> is the [[quantile function]] associated with ''F''. One of the first people to mention and prove this result was [[Frederick Mosteller]] in his seminal paper in 1946.<ref name = "Mosteller">{{cite journal|last = Mosteller| first = Frederick| author-link = Frederick Mosteller| year = 1946| title = On Some Useful "Inefficient" Statistics| url = http://projecteuclid.org/euclid.aoms/1177730881| journal = [[Annals of Mathematical Statistics]]| volume = 17| issue = 4| pages = 377–408| doi = 10.1214/aoms/1177730881| access-date = February 26, 2015| doi-access = free}}</ref> Further research led in the 1960s to the [[Raghu Raj Bahadur|Bahadur]] representation which provides information about the errorbounds. The convergence to normal distribution also holds in a stronger sense, such as convergence in [[Kullback–Leibler divergence|relative entropy or KL divergence]].<ref>M. Cardone, A. Dytso and C. Rush, "Entropic Central Limit Theorem for Order Statistics," in IEEE Transactions on Information Theory, vol. 69, no. 4, pp. 2193-2205, April 2023, doi: 10.1109/TIT.2022.3219344.</ref>

An interesting observation can be made in the case where the distribution is symmetric, and the population median equals the population mean. In this case, the [[sample mean]], by the [[central limit theorem]], is also asymptotically normally distributed, but with variance σ<sup>2</sup>''/n'' instead. This asymptotic analysis suggests that the mean outperforms the median in cases of low [[kurtosis]], and vice versa. For example, the median achieves better confidence intervals for the [[Laplace distribution]], while the mean performs better for ''X'' that are normally distributed.

==== Proof ====
It can be shown that

: <math>B(k,n+1-k)\ \stackrel{\mathrm{d}}{=}\ \frac{X}{X + Y},</math>

where

: <math> X = \sum_{i=1}^{k} Z_i, \quad Y = \sum_{i=k+1}^{n+1} Z_i,</math>

with ''Z<sub>i</sub>'' being independent identically distributed [[exponential distribution|exponential]] random variables with rate 1. Since ''X''/''n'' and ''Y''/''n'' are asymptotically normally distributed by the CLT, our results follow by application of the [[delta method]].