Editing Order statistic (section)

==== Order statistics sampled from a uniform distribution ====

In this section we show that the order statistics of the [[uniform distribution (continuous)|uniform distribution]] on the [[unit interval]] have [[marginal distribution]]s belonging to the [[beta distribution]] family. We also give a simple method to derive the joint distribution of any number of order statistics, and finally translate these results to arbitrary continuous distributions using the [[cumulative distribution function|cdf]].

We assume throughout this section that <math>X_1, X_2, \ldots, X_n</math> is a [[random sample]] drawn from a continuous distribution with cdf <math>F_X</math>. Denoting <math>U_i=F_X(X_i)</math> we obtain the corresponding random sample <math>U_1,\ldots,U_n</math> from the standard [[uniform distribution (continuous)|uniform distribution]]. Note that the order statistics also satisfy <math>U_{(i)}=F_X(X_{(i)})</math>.

The probability density function of the order statistic <math>U_{(k)}</math> is equal to<ref name="gentle">{{citation|title=Computational Statistics|first=James E.|last=Gentle|publisher=Springer|year=2009|isbn=9780387981444|page=63|url=https://books.google.com/books?id=mQ5KAAAAQBAJ&pg=PA63}}.</ref>

:<math>f_{U_{(k)}}(u)={n!\over (k-1)!(n-k)!}u^{k-1}(1-u)^{n-k}</math>

that is, the ''k''th order statistic of the uniform distribution is a [[beta distribution|beta-distributed]] random variable.<ref name="gentle"/><ref>{{citation|title=Kumaraswamy's distribution: A beta-type distribution with some tractability advantages|first=M. C.|last=Jones|journal=Statistical Methodology|volume=6|issue=1|year=2009|pages=70–81|doi=10.1016/j.stamet.2008.04.001|quote=As is well known, the beta distribution is the distribution of the ''m'' ’th order statistic from a random sample of size ''n'' from the uniform distribution (on (0,1)).}}</ref>

:<math>U_{(k)} \sim \operatorname{Beta}(k,n+1\mathbf{-}k).</math>

The proof of these statements is as follows. For <math>U_{(k)}</math> to be between ''u'' and ''u''&nbsp;+&nbsp;''du'', it is necessary that exactly ''k''&nbsp;−&nbsp;1 elements of the sample are smaller than ''u'', and that at least one is between ''u'' and ''u''&nbsp;+&nbsp;d''u''. The probability that more than one is in this latter interval is already <math>O(du^2)</math>, so we have to calculate the probability that exactly ''k''&nbsp;−&nbsp;1, 1 and ''n''&nbsp;−&nbsp;''k'' observations fall in the intervals <math>(0,u)</math>, <math>(u,u+du)</math> and <math>(u+du,1)</math> respectively. This equals (refer to [[multinomial distribution]] for details)

:<math>{n!\over (k-1)!(n-k)!}u^{k-1}\cdot du\cdot(1-u-du)^{n-k}</math>

and the result follows.

The mean of this distribution is ''k'' / (''n'' + 1).