Order Statistics Uniform Pop.| Real Statistics Using Excel

Basic Properties

The following properties are Properties 3 and 4 from Order Statistics from a Continuous Uniform Distribution. The proofs are shown there.

Property 1: If the population follows the uniform distribution on the interval (0,1), the kth order statistic has a beta distribution Bet(k, n-k+1).

Property 2: If the population follows the uniform distribution on the interval (0,1), the mean and variance of the kth order statistic are

E.g. the mean of the 3^rd order statistic from a sample of size 10 is 3/11 = .2727. We get the same value when using the formula =ORDER_MEAN(3,10,,”uniform”,0,1). See Order Statistics from a Continuous Population.

Observation: For a sample of size n from a population with a uniform distribution on (0,1), the distribution of the kth order statistic can be expressed in Excel by

f_k(x) = BETA.DIST(x,k,n-k+1,FALSE)

F_k(x) = BETA.DIST(x,k,n-k+1,TRUE)

Generalization

For any continuous distribution, the cdf F(x) is an increasing function on the interval (0,1). Thus, if x₁, …, x_n is a sample from this distribution in increasing order, then F(x₁), …, F(x_n) is a sample from the uniform distribution on (0,1) and the kth order statistic from the uniform distribution is F(x_(k)).

Property 3: For any distribution with cdf F(x)

F_k(x) = BETA.DIST(F(x),k,n-k+1,TRUE)

Proof: F(x) has a uniform distribution on (0,1) and so

F(x)∼ Bet(k, n-k+1)

and the result follows from Property 1.

Observation: For example, the kth order statistic for the gamma distribution Gamma(α, β) could be calculated in Excel by the formula

F_k(x) = BETA.DIST(GAMMA.DIST(x,α,β,TRUE),k,n-k+1,TRUE)

Joint Distribution

Property 4: For a uniform distribution on (0,1) where x < y

Proof: By Property 3 of Joint and Range Order Statistics from a Continuous Population

Range

Property 5: For a uniform distribution on (0,1)

Proof: Click here for the proof using calculus.

Observation: Note that the distribution of x_(k) – x_(j) only depends on the difference k – j. Thus, for example, the distribution of x₍₅₎ – x₍₂₎ is the same as the distribution of x₍₇₎ – x₍₄₎.