Order Statistics

In sampling statistics (see Sec 18.7), we deal with an iid class of random variables, where n is a prescribed positive integer known as the sample size. An observation of this class gives an n-tuple of numbers . As an extension of the extreme values in the case of two variables (Ex 2.11), it is often useful to define a random variable Y1 whose value for any ω is the smallest of the X i (ω); a second random variable Y2 whose value at ω is the next smallest of the X i (ω), and so on through Yn whose value at ω is the largest of the X i (ω). We would like to be able to obtain the distributions for these new random variables in terms of the common distribution for the Xi . We formulate the problem as follows.

Example 1. Order Statistics

Suppose is iid, with common distribution function F. Let

  • smallest of

  • next larger of

  • . . .

  • largest of

Then Yk is called the kth order statistic for the class . We wish to determine the distribution functions . Now, iff k or more of the Xi have values no greater than t. We may view the process as a Bernoulli sequence of n trials. There is a success on the ith trial iff . The probability p of a success is . Hence

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Remark. Once the common distribution function F for the Xi is known, then the Fk are calculated in a straightforward manner. For that purpose we may use the MATLAB function cbinom.

Example 2. 

Suppose the Xi are exponential (2). Then F X (t) = 1 – e – 2t for positive t. Suppose n = 5. We calculate F k (t) for . 

n = 5;
t = 0.1:0.2:0.9;
m = length(t);
F = 1 - exp(-2*t);
for i = 1:m
FK(i,:) = cbinom(n,F(i),1:n);
end
disp([t' F' FK])    % k = 1    k = 2     k = 3     k = 4     k = 5
0.1000    0.1813    0.6321    0.2249    0.0445    0.0046    0.0002
0.3000    0.4512    0.9502    0.7456    0.4091    0.1324    0.0187
0.5000    0.6321    0.9933    0.9354    0.7364    0.3946    0.1009
0.7000    0.7534    0.9991    0.9852    0.9000    0.6400    0.2427
0.9000    0.8347    0.9999    0.9968    0.9653    0.8064    0.4052

The following special case is important in characterizing the Poisson process (see Sec 21.1).

Example 3. 

Order statistics for uniformly distributed random variables

Suppose is iid, uniform on (0,T]. Determine the distribution functions for the order statistics.

SOLUTION

The common distribution function for the Ui is given by . According to the result in Ex 2.16, the kth order statistic Yk has the distribution function

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