Order Statistics Proofs

Range statistics from a continuous population

Property 6 from Range Statistics from a Continuous Population

The cdf of w = x(n) – x(1) is equal to

Cdf of range statistic

Proof: By Property 5 of Range Statistics from a Continuous Population

Proof 1

Proof 2

Proof 3

Proof 4

Observation: The cdf G(w) of the range w = x(n)x(1) is

Change of variables

This results from the above property using the substitution u = F(x). Then x = F-1(u) and du = f(x)dx.

The same change in variables for Property 5 of Range Statistics from a Continuous Population yields

Pdf change of variables

Order statistics from a uniform distribution

Property 5 from Order Statistics from a Uniform Distribution

For a uniform distribution on (0,1)

Proof 2

Proof: Assume that 0 < x < y < 1 and w = y-x. Thus, 0 < x < w+x < 1. Using the fact that y = w+x, by Property 4 of Order Statistics from a Uniform Distribution and Property 5 from Range Statistics from a Continuous Population, we see that the pdf g(w) of x(k)x(j) is

Proof 1

Property 2

We now use the substitution x = z(1-w) and so dx = (1-w)dz and z = 1 when x = 1-w

Proof 3

Proof 4

Proof 5

Thus

Proof 6

But

Proof 7

Hence

Proof 8

Proof 9

Proof 10

Proof 11

which is the pdf of Bet(k-j, n-(k-j)+1).

References

David, H. A. and Nagaraja, H. N. (2003) Order statistics. Wiley
https://books.google.it/books/about/Order_Statistics.html?id=3Ts1yDLWXmQC&redir_esc=y

Omondi, O. C. (2016) Order statistics of uniform, logistic and exponential distributions
http://erepository.uonbi.ac.ke/bitstream/handle/11295/97307/MSc_Project2016.pdf?sequence=1&isAllowed=y

Arnold, B. C., Balakrishnan, N., Nagaraja, H. N. (2003) A First course in order statistics. Society for Industrial and Applied Mathematics
https://books.google.it/books/about/A_First_Course_in_Order_Statistics.html?id=gUD-S8USlDwC&redir_esc=y

Leave a Comment