Order Statistic

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.

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.