In a sample taken from a continuous population, the kth order statistic is the kth smallest element in the sample. If we assume that the order of the elements in the sample is x1 < x2 < … < xn, then the kth order statistic, denoted x(k) is xk. Just as the mean can be treated as a random variable, we will also use the notation x(k) to represent a random variable.
Properties
Suppose that the population has a continuous distribution with pdf f(u) and cdf F(u). First, we describe the (cumulative) distribution Fk(x) of the kth order statistic in a sample of size n taken from the population.
Property 1: The cdf of the kth order statistic in the sample is
Proof:
The probability that i elements in the sample are less than or equal to x is equivalent to tossing a coin n times and getting i heads where the probability of heads on any toss is F(x). Using the binomial distribution this is C(n,i)F(x)n[1-F(x)]n-i, which completes the proof.
Observation: If n is odd so that n = 2k+1, the cdf of the kth order statistic (the median) in the sample is
Observation: The cdf of the nth-order statistic is
Alternatively,
Similarly, the cdf of the first-order statistic
Observation: Using algebra, we see that the inverse functions for the first- and last-order statistics are:
Property 2: The pdf of the kth order statistic is
Observation: We have the special cases
Worksheet Functions
Real Statistics Functions: The Real Statistics Resource Pack supports the following worksheet functions. These functions refer to a distribution dist (“uniform”, “normal”, etc.) with the specified parameters as described for the MEAN_DIST and VAR_DIST functions (see Distribution Property Functions).
ORDER_DIST(x, k, n, cum, dist, param1, param2, param3) = the pdf f(x) for the kth order statistic from a sample of size n for the specified distribution if cum = FALSE and the corresponding cdf F(x) if cum = TRUE.
ORDER_INV(p, k, n, dist, param1, param2, param3) = the inverse at p for the kth order statistic from a sample of size n for the specified distribution; i.e. the value x such that F(x) = p.
ORDER_MEAN(k, n, dist, param1, param2, param3) = the expected value of the kth order statistic from a sample of size n for the specified distribution.
Examples
For example, =ORDER_DIST(7,5,11,TRUE,”laplace”,10,5) takes the value .1576. Thus for a random sample of size 11 taken from the Laplace distribution with mu = 10 and beta = 5, the cdf F5(7) = .157588 for the 5th order statistic. This means that the probability that 5th order statistic from a sample of size 11 from the specified Laplace distribution is less than or equal to 7 is 15.76%.
Thus, =ORDER_INV(.157588,5,11,”laplace”,10,5) takes the value 7. Note, however, that =ORDER_INV(.1576,5,11,”laplace”,10,5) takes the value 8.
The value of =ORDER_MEAN(6,11,,”laplace”,10,5) is 10. This means that the expected value for the median, i.e. the 6th order statistic, of the Laplace distribution with μ = 10 and β = 5 is 10. This is as expected since in general μ is the median of a Laplace distribution.
Examples Workbook
Click here to download the Excel workbook with the examples described on this webpage.
References
Chen, P-N (2008) Basic theories on order statistics
Reference no longer available
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
Ma D. (2010) The distribution of the order statistics. A Blog on probability and statistics
https://probabilityandstats.wordpress.com/2010/02/20/the-distributions-of-the-order-statistics/
Border, K. C. (2016) Lecture 15: Order statistics; conditional expectation. Caltech
https://healy.econ.ohio-state.edu/kcb/Ma103/Notes/Lecture15.pdf