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)
midrange
(4)
and statistical median
(5)
(Hogg and Craig 1970, p. 152).
If
has probability density function and distribution
function ,
then the probability function of is given by
(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." §9.4 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. Referenced on Wolfram|Alpha Order Statistic
Cite this as:
Weisstein, Eric W. "Order Statistic."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/OrderStatistic.html
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