Given a sample of 
variates 
,
..., 
,
reorder them so that 
.
Then 
is called the 
th
order statistic (Hogg and Craig 1970, p. 146), sometimes also denoted 
. Special cases include the minimum
 |
(1) |
and maximum
 |
(2) |
Important functions of order statistics include the statistical range
 |
(3) |
 |
(4) |
 |
(5) |
(Hogg and Craig 1970, p. 152).
If 
has probability density function 
and distribution
function 
,
then the probability function of 
is given by
![f_(Y_r)=(N!)/((r-1)!(N-r)!)[F(x)]^(r-1)[1-F(x)]^(N-r)f(x)](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FOrderStatistic%2FNumberedEquation6.svg){kind=small} |
(6) |
for 
,
..., 
(Rose and Smith 2002, pp. 311 and 454).
A robust estimation technique based on linear combinations of order statistics is called an L-estimate.
See also
Extreme Value Distribution, Hinge, Maximum, Midrange, Minimum, Statistical Median
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References
Balakrishnan, N. and Chen, W. W. S. Handbook of Tables for Order Statistics from Lognormal Distributions with Applications. Amsterdam, Netherlands: Kluwer, 1999.Balakrishnan, N. and Cohen, A. C. Order Statistics and Inference. New York: Academic Press, 1991.Balakrishnan, N. and Rao, C. R. (Eds.). Handbook of Statistics, Vol. 16: Order Statistics: Theory and Methods. Amsterdam, Netherlands: Elsevier, 1998.Balakrishnan, N. and Rao, C. R. (Eds.). Order Statistics: Applications. Amsterdam, Netherlands: Elsevier, 1998.David, H. A. Order Statistics, 2nd ed. New York: Wiley, 1981.Gibbons, J. D. and Chakraborti, S. (Eds.). Nonparametric Statistic Inference, 3rd ed. exp. rev. New York: Dekker, 1992.Hogg, R. V. and Craig, A. T. Introduction to Mathematical Statistics, 3rd ed. New York: Macmillan, 1970.Rose, C. and Smith, M. D. "Order Statistics." ยง9.4 in Mathematical Statistics with Mathematica. New York: Springer-Verlag, pp. 311-322, 2002.Rose, C. and Smith, M. D. "Computational Order Statistics." Mathematica J. 9, 790-802, 2005.
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Cite this as:
Weisstein, Eric W. "Order Statistic." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/OrderStatistic.html