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Order Statistic

Given a sample of n variates X_1, ..., X_N, reorder them so that Y_1<Y_2<...<Y_N. Then Y_i is called the ith order statistic (Hogg and Craig 1970, p. 146), sometimes also denoted X^(<i>). Special cases include the minimum

Y_1=min_(j)(X_j)
(1)

and maximum

Y_N=max_(j)(X_j).
(2)

Important functions of order statistics include the statistical range

R=Y_N-Y_1,
(3)

midrange

MR=1/2(Y_1+Y_N),
(4)

and statistical median

x^~={Y_((N+1)/2) if N is odd; 1/2(Y_(N/2)+Y_(1+N/2)) if N is even
(5)

(Hogg and Craig 1970, p. 152).

If X has probability density function f(x) and distribution function F(x), then the probability function of Y_r is given by

f_(Y_r)=(N!)/((r-1)!(N-r)!)[F(x)]^(r-1)[1-F(x)]^(N-r)f(x)
(6)

for r=1, ..., N (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.

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Order Statistic

Cite this as:

Weisstein, Eric W. "Order Statistic." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/OrderStatistic.html

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