Order Statistics /joint pdf of order statistics / pdf of order statistics / joint pdf of two order statistics /pdf and cdf of first and the largest order statistics /median/sample range/ Relation between U(0,1) andBeta

<Order Statistics>

Order statistics are random samples listed in increasing order.

I think you might be familiar with the minimum, maximum, median, and quantiles which are also the order statistics. 

X(1),X(2),X(3),...,X(n) are order statistics and X(k) is called as 'k'th order statistics. 

<Joint PDF of order statistics >

Joint probability density function of order statistics is the pdf of X(1)<=X(2)<=X(3)<=...<X(n) which is f(x(1),x(2),..,x(n))

There are n cases for being X(1) because there are n random samples and as X(1) is fixed, there are n-1 cases for being X(2), and so on. Finally, only a case left for X(n) which is the largest order statistics. ----->n(n-1)(n-2)..(1)--->n!

so pdf of order statistics is n!f(x(1),x(2),x(3),...x(n)).

<PDF of 'k'th order statistics X(k)>

X(k) is called as 'k'th order statistic which is 'k'th largest number of samples. 

There are k-1 samples that are smaller than X(k) and n-k samples which are larger than X(k) in n samples. We simply regard every order statistics in three ways. 'k'th order statistics, smaller than 'k'th order statistics (1-F(X(k)), and 'larger than 'k'th order statistics' F(X(k)). -------------> F(X(k))^(k-1) * (1-F(X(k))^(n-k)) * f(X(k))

and there are n!/(k-1)!(n-k)! ways to select X(k) because there are (k-1) smaller statistics, (n-k) larger statistics. 

 

<Joint PDF of two order statistics, X(i), X(j)>

A similar way to find the pdf for X(k).

<CDF and PDF of first order statistics, X(1)>

First order statistics, X(1) which is 'minimum' is one of the most frequently used order statistics. If you want to find pdf of the first order statistics, you must find CDF of X(1). 

'probability that X(1) is smaller than x'

='1-probability that X(1) is greater than x'

='1-probability that every order statistics are greater than x'

='1-(1-F(x))^n'

and you can find PDF of X(1) by differentiating CDF by X(1)

<CDF and PDF of nth order statistics, X(n)>

X(n), maximum, is one of the most frequently used order statistics. If you want to find pdf of the largest order statistics, you must find CDF of X(n).

'probability that X(n) is smaller than x'

='probability that every order statistics are smaller than x'

='(F(x)^n)'

and you can find PDF of X(n) by differentiating CDF by X(n)

<Example- PDF of X(1),X(n) of exponential distribution>

<definition of median> 

Median(X) is middle number from samples. 

If the number of samples is n, median(x) should be the middle number from n samples. 

If there are odd numbers of samples, median becomes middle number from n samples with same numbers below and above. 

If there are even numbers of samples, median becomes mean of middle pair numbers. 

<Sample Range>

Sample range is an important part of order statistics. From this, you can know the spread of the samples from the smallest number to the largest number. 

<Special Case :  Relationship between Standard Uniform distribution - Beta distribution>

Kth order statistics of standard uniform distribution becomes the same form with Beta function 

with parameter k and n+1-k. 

X(k) of U(0,1) == Beta (k, n+1-k)