Probability density functions of the order statistics for a sample of size n = 5 from an exponential distribution with unit scale parameter

In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value.^[1] Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference.

Important special cases of the order statistics are the minimum and maximum value of a sample, and (with some qualifications discussed below) the sample median and other sample quantiles.

When using probability theory to analyze order statistics of random samples from a continuous distribution, the cumulative distribution function is used to reduce the analysis to the case of order statistics of the uniform distribution.

Notation and examples

For example, suppose that four numbers are observed or recorded, resulting in a sample of size 4. If the sample values are

6, 9, 3, 8,

the order statistics would be denoted

${\displaystyle x_{(1)}=3,\ \ x_{(2)}=6,\ \ x_{(3)}=8,\ \ x_{(4)}=9,\,}$

where the subscript (i) enclosed in parentheses indicates the ith order statistic of the sample.

The first order statistic (or smallest order statistic) is always the minimum of the sample, that is,

${\displaystyle X_{(1)}=\min\{\,X_{1},\ldots ,X_{n}\,\}}$

where, following a common convention, we use upper-case letters to refer to random variables, and lower-case letters (as above) to refer to their actual observed values.

Similarly, for a sample of size n, the nth order statistic (or largest order statistic) is the maximum, that is,

${\displaystyle X_{(n)}=\max\{\,X_{1},\ldots ,X_{n}\,\}.}$

The sample range is the difference between the maximum and minimum. It is a function of the order statistics:

${\displaystyle {\rm {Range}}\{\,X_{1},\ldots ,X_{n}\,\}=X_{(n)}-X_{(1)}.}$

A similar important statistic in exploratory data analysis that is simply related to the order statistics is the sample interquartile range.

The sample median may or may not be an order statistic, since there is a single middle value only when the number n of observations is odd. More precisely, if n = 2m+1 for some integer m, then the sample median is ${\displaystyle X_{(m+1)}}$ and so is an order statistic. On the other hand, when n is even, n = 2m and there are two middle values, ${\displaystyle X_{(m)}}$ and ${\displaystyle X_{(m+1)}}$, and the sample median is some function of the two (usually the average) and hence not an order statistic. Similar remarks apply to all sample quantiles.

Probabilistic analysis

Given any random variables X₁, X₂..., X_n, the order statistics X₍₁₎, X₍₂₎, ..., X_(n) are also random variables, defined by sorting the values (realizations) of X₁, ..., X_n in increasing order.

When the random variables X₁, X₂..., X_n form a sample they are independent and identically distributed. This is the case treated below. In general, the random variables X₁, ..., X_n can arise by sampling from more than one population. Then they are independent, but not necessarily identically distributed, and their joint probability distribution is given by the Bapat–Beg theorem.

From now on, we will assume that the random variables under consideration are continuous and, where convenient, we will also assume that they have a probability density function (PDF), that is, they are absolutely continuous. The peculiarities of the analysis of distributions assigning mass to points (in particular, discrete distributions) are discussed at the end.

Probability distributions of order statistics

Order statistics sampled from a uniform distribution

In this section we show that the order statistics of the uniform distribution on the unit interval have marginal distributions 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 cdf.

We assume throughout this section that ${\displaystyle X_{1},X_{2},\ldots ,X_{n}}$ is a random sample drawn from a continuous distribution with cdf ${\displaystyle F_{X}}$. Denoting ${\displaystyle U_{i}=F_{X}(X_{i})}$ we obtain the corresponding random sample ${\displaystyle U_{1},\ldots ,U_{n}}$ from the standard uniform distribution. Note that the order statistics also satisfy ${\displaystyle U_{(i)}=F_{X}(X_{(i)})}$.

The probability density function of the order statistic ${\displaystyle U_{(k)}}$ is equal to^[3]

${\displaystyle f_{U_{(k)}}(u)={n! \over (k-1)!(n-k)!}u^{k-1}(1-u)^{n-k}}$

that is, the kth order statistic of the uniform distribution is a beta-distributed random variable.^[3][4]

${\displaystyle U_{(k)}\sim \operatorname {Beta} (k,n+1\mathbf {-} k).}$

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

${\displaystyle {n! \over (k-1)!(n-k)!}u^{k-1}\cdot du\cdot (1-u-du)^{n-k}}$

and the result follows.

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

The joint distribution of the order statistics of the uniform distribution

Similarly, for i < j, the joint probability density function of the two order statistics U_(i) < U_(j) can be shown to be

${\displaystyle f_{U_{(i)},U_{(j)}}(u,v)=n!{u^{i-1} \over (i-1)!}{(v-u)^{j-i-1} \over (j-i-1)!}{(1-v)^{n-j} \over (n-j)!}}$

which is (up to terms of higher order than ${\displaystyle O(du\,dv)}$) the probability that i − 1, 1, j − 1 − i, 1 and n − j sample elements fall in the intervals ${\displaystyle (0,u)}$, ${\displaystyle (u,u+du)}$, ${\displaystyle (u+du,v)}$, ${\displaystyle (v,v+dv)}$, ${\displaystyle (v+dv,1)}$ respectively.

One reasons in an entirely analogous way to derive the higher-order joint distributions. Perhaps surprisingly, the joint density of the n order statistics turns out to be constant:

${\displaystyle f_{U_{(1)},U_{(2)},\ldots ,U_{(n)}}(u_{1},u_{2},\ldots ,u_{n})=n!.}$

One way to understand this is that the unordered sample does have constant density equal to 1, and that there are n! different permutations of the sample corresponding to the same sequence of order statistics. This is related to the fact that 1/n! is the volume of the region ${\displaystyle 0<u_{1}<\cdots <u_{n}<1}$. It is also related with another particularity of order statistics of uniform random variables: It follows from the BRS-inequality that the maximum expected number of uniform U(0,1] random variables one can choose from a sample of size n with a sum up not exceeding ${\displaystyle 0<s<n/2}$ is bounded above by ${\displaystyle {\sqrt {2sn}}}$, which is thus invariant on the set of all ${\displaystyle s,n}$ with constant product ${\displaystyle sn}$.

Using the above formulas, one can derive the distribution of the range of the order statistics, that is the distribution of ${\displaystyle U_{(n)}-U_{(1)}}$, i.e. maximum minus the minimum. More generally, for ${\displaystyle n\geq k>j\geq 1}$, ${\displaystyle U_{(k)}-U_{(j)}}$ also has a beta distribution:

$${\displaystyle U_{(k)}-U_{(j)}\sim \operatorname {Beta} (k-j,n-(k-j)+1)}$$

From these formulas we can derive the covariance between two order statistics:

$${\displaystyle \operatorname {Cov} (U_{(k)},U_{(j)})={\frac {j(n-k+1)}{(n+1)^{2}(n+2)}}}$$

The formula follows from noting that

$${\displaystyle \operatorname {Var} (U_{(k)}-U_{(j)})=\operatorname {Var} (U_{(k)})+\operatorname {Var} (U_{(j)})-2\cdot \operatorname {Cov} (U_{(k)},U_{(j)})={\frac {k(n-k+1)}{(n+1)^{2}(n+2)}}+{\frac {j(n-j+1)}{(n+1)^{2}(n+2)}}-2\cdot \operatorname {Cov} (U_{(k)},U_{(j)})}$$

and comparing that with

$${\displaystyle \operatorname {Var} (U)={\frac {(k-j)(n-(k-j)+1)}{(n+1)^{2}(n+2)}}}$$

where ${\displaystyle U\sim \operatorname {Beta} (k-j,n-(k-j)+1)}$, which is the actual distribution of the difference.

Order statistics sampled from an exponential distribution

For ${\displaystyle X_{1},X_{2},..,X_{n}}$ a random sample of size n from an exponential distribution with parameter λ, the order statistics X_(i) for i = 1,2,3, ..., n each have distribution

${\displaystyle X_{(i)}{\stackrel {d}{=}}{\frac {1}{\lambda }}\left(\sum _{j=1}^{i}{\frac {Z_{j}}{n-j+1}}\right)}$

where the Z_j are iid standard exponential random variables (i.e. with rate parameter 1). This result was first published by Alfréd Rényi.^[5][6]

Order statistics sampled from an Erlang distribution

The Laplace transform of order statistics may be sampled from an Erlang distribution via a path counting method .^[7]

The joint distribution of the order statistics of an absolutely continuous distribution

If F_X is absolutely continuous, it has a density such that ${\displaystyle dF_{X}(x)=f_{X}(x)\,dx}$, and we can use the substitutions

${\displaystyle u=F_{X}(x)}$

and

${\displaystyle du=f_{X}(x)\,dx}$

to derive the following probability density functions for the order statistics of a sample of size n drawn from the distribution of X:

${\displaystyle f_{X_{(k)}}(x)={\frac {n!}{(k-1)!(n-k)!}}[F_{X}(x)]^{k-1}[1-F_{X}(x)]^{n-k}f_{X}(x)}$

${\displaystyle f_{X_{(j)},X_{(k)}}(x,y)={\frac {n!}{(j-1)!(k-j-1)!(n-k)!}}[F_{X}(x)]^{j-1}[F_{X}(y)-F_{X}(x)]^{k-1-j}[1-F_{X}(y)]^{n-k}f_{X}(x)f_{X}(y)}$ where ${\displaystyle x\leq y}$

${\displaystyle f_{X_{(1)},\ldots ,X_{(n)}}(x_{1},\ldots ,x_{n})=n!f_{X}(x_{1})\cdots f_{X}(x_{n})}$ where ${\displaystyle x_{1}\leq x_{2}\leq \dots \leq x_{n}.}$

Application: confidence intervals for quantiles

An interesting question is how well the order statistics perform as estimators of the quantiles of the underlying distribution.

Application: Non-parametric density estimation

Moments of the distribution for the first order statistic can be used to develop a non-parametric density estimator.^[10] Suppose, we want to estimate the density ${\displaystyle f_{X}}$ at the point ${\displaystyle x^{*}}$. Consider the random variables ${\displaystyle Y_{i}=|X_{i}-x^{*}|}$, which are i.i.d with distribution function ${\displaystyle g_{Y}(y)=f_{X}(y+x^{*})+f_{X}(x^{*}-y)}$. In particular, ${\displaystyle f_{X}(x^{*})={\frac {g_{Y}(0)}{2}}}$.

The expected value of the first order statistic ${\displaystyle Y_{(1)}}$ given a sample of ${\displaystyle N}$ total observations yields,

${\displaystyle E(Y_{(1)})={\frac {1}{(N+1)g(0)}}+{\frac {1}{(N+1)(N+2)}}\int _{0}^{1}Q''(z)\delta _{N+1}(z)\,dz}$

where ${\displaystyle Q}$ is the quantile function associated with the distribution ${\displaystyle g_{Y}}$, and ${\displaystyle \delta _{N}(z)=(N+1)(1-z)^{N}}$. This equation in combination with a jackknifing technique becomes the basis for the following density estimation algorithm,

  Input: A sample of ${\displaystyle N}$ observations. ${\displaystyle \{x_{\ell }\}_{\ell =1}^{M}}$ points of density evaluation. Tuning parameter ${\displaystyle a\in (0,1)}$ (usually 1/3).
  Output: ${\displaystyle \{{\hat {f}}_{\ell }\}_{\ell =1}^{M}}$ estimated density at the points of evaluation.

  1: Set ${\displaystyle m_{N}=\operatorname {round} (N^{1-a})}$
  2: Set ${\displaystyle s_{N}={\frac {N}{m_{N}}}}$
  3: Create an ${\displaystyle s_{N}\times m_{N}}$ matrix ${\displaystyle M_{ij}}$ which holds ${\displaystyle m_{N}}$ subsets with ${\displaystyle s_{N}}$ observations each.
  4: Create a vector ${\displaystyle {\hat {f}}}$ to hold the density evaluations.
  5: for ${\displaystyle \ell =1\to M}$ do
  6:     for ${\displaystyle k=1\to m_{N}}$ do
  7:         Find the nearest distance ${\displaystyle d_{\ell k}}$ to the current point ${\displaystyle x_{\ell }}$ within the ${\displaystyle k}$th subset
  8:      end for
  9:      Compute the subset average of distances to ${\displaystyle x_{\ell }:d_{\ell }=\sum _{k=1}^{m_{N}}{\frac {d_{\ell k}}{m_{N}}}}$
 10:      Compute the density estimate at ${\displaystyle x_{\ell }:{\hat {f}}_{\ell }={\frac {1}{2(1+s_{N})d_{\ell }}}}$
 11:  end for
 12: return ${\displaystyle {\hat {f}}}$

In contrast to the bandwidth/length based tuning parameters for histogram and kernel based approaches, the tuning parameter for the order statistic based density estimator is the size of sample subsets. Such an estimator is more robust than histogram and kernel based approaches, for example densities like the Cauchy distribution (which lack finite moments) can be inferred without the need for specialized modifications such as IQR based bandwidths. This is because the first moment of the order statistic always exists if the expected value of the underlying distribution does, but the converse is not necessarily true.^[11]

Dealing with discrete variables

Suppose ${\displaystyle X_{1},X_{2},\ldots ,X_{n}}$ are i.i.d. random variables from a discrete distribution with cumulative distribution function ${\displaystyle F(x)}$ and probability mass function ${\displaystyle f(x)}$. To find the probabilities of the ${\displaystyle k^{\text{th}}}$ order statistics, three values are first needed, namely

${\displaystyle p_{1}=P(X<x)=F(x)-f(x),\ p_{2}=P(X=x)=f(x),{\text{ and }}p_{3}=P(X>x)=1-F(x).}$

The cumulative distribution function of the ${\displaystyle k^{\text{th}}}$ order statistic can be computed by noting that

${\displaystyle {\begin{aligned}P(X_{(k)}\leq x)&=P({\text{there are at least }}k{\text{ observations less than or equal to }}x),\\&=P({\text{there are at most }}n-k{\text{ observations greater than }}x),\\&=\sum _{j=0}^{n-k}{n \choose j}p_{3}^{j}(p_{1}+p_{2})^{n-j}.\end{aligned}}}$

Similarly, ${\displaystyle P(X_{(k)}<x)}$ is given by

${\displaystyle {\begin{aligned}P(X_{(k)}<x)&=P({\text{there are at least }}k{\text{ observations less than }}x),\\&=P({\text{there are at most }}n-k{\text{ observations greater than or equal to }}x),\\&=\sum _{j=0}^{n-k}{n \choose j}(p_{2}+p_{3})^{j}(p_{1})^{n-j}.\end{aligned}}}$

Note that the probability mass function of ${\displaystyle X_{(k)}}$ is just the difference of these values, that is to say

${\displaystyle {\begin{aligned}P(X_{(k)}=x)&=P(X_{(k)}\leq x)-P(X_{(k)}<x),\\&=\sum _{j=0}^{n-k}{n \choose j}\left(p_{3}^{j}(p_{1}+p_{2})^{n-j}-(p_{2}+p_{3})^{j}(p_{1})^{n-j}\right),\\&=\sum _{j=0}^{n-k}{n \choose j}\left((1-F(x))^{j}(F(x))^{n-j}-(1-F(x)+f(x))^{j}(F(x)-f(x))^{n-j}\right).\end{aligned}}}$

Computing order statistics

The problem of computing the kth smallest (or largest) element of a list is called the selection problem and is solved by a selection algorithm. Although this problem is difficult for very large lists, sophisticated selection algorithms have been created that can solve this problem in time proportional to the number of elements in the list, even if the list is totally unordered. If the data is stored in certain specialized data structures, this time can be brought down to O(log n). In many applications all order statistics are required, in which case a sorting algorithm can be used and the time taken is O(n log n).

See also

• Rankit
• Box plot
• BRS-inequality
• Concomitant (statistics)
• Fisher–Tippett distribution
• Bapat–Beg theorem for the order statistics of independent but not necessarily identically distributed random variables
• Bernstein polynomial
• L-estimator – linear combinations of order statistics
• Rank-size distribution
• Selection algorithm

References

External links

• Order statistics at PlanetMath. Retrieved Feb 02, 2005
• Weisstein, Eric W. "Order Statistic". MathWorld. Retrieved Feb 02, 2005
• C++ source Dynamic Order Statistics