Definition of order statistics

Let X be a random variable with p(x) its (continuous) probability density function. If several n observation samples are drawn from this distribution, the position of the leftmost observation will vary from sample to sample: it is a random variable.

More generally, for a given n, the random variable X_(k) is defined as the value of the k^th observation from the left, and is called the "k^th order statistics" of the original variable X.

Examples of order statistics

Special cases of order statistics that are important in practice are:

* The smallest (leftmost) observation X₍₁₎.

* The largest (rightmost) observation X_(n).

Important functions of order statistics are:

* The median, defined as:

X_(n+1)/2 if n is odd,
(X_n/2 + X_(n+1)/2 )/2 if n is even.

* The sample range, defined as the distance between the two extreme observations:

X_(n) - X₍₁₎

* The sample mid-range, defined as:

(X₍₁₎ + X_(n)) / 2

and which is a robust estimate of the central tendency of the distribution of X.

Animation

You'll find here an interactive animation that illustates the distribution of the order statistics of the uniform distribution.

Estimation of q of the uniform distribution U[0, q]

The rightmost observation of a sample drawn form U[0, q] :

* Is a sufficient statistic for the parameter q.

* Can be used for building an unbiased estimator of q.

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Tutorial

In this Tutorial, we derive the probability density function of the order statistics of the [0, q] uniform distribution. We do it two different ways:

* First by using an intuitive, half-rigourous argument.

* Then by differentiating the distribution function of the order statistics. This approach is more rigorous but does not appeal nearly as much to the imagination.

In both approaches, the binomial distribution B(n, p) plays a central role.

DISTRIBUTIONS OF THE ORDER STATISTICS

OF THE UNIFORM DISTRIBUTION

Uniform distribution [0, 1]

The intuitive approach

The distribution function approach

The distribution function

The probability density function

Uniform distribution [0, q]

General case

Extreme observtions

Leftmost observation