Order Statistic
المؤلف:
Balakrishnan, N. and Chen, W. W. S.
المصدر:
Handbook of Tables for Order Statistics from Lognormal Distributions with Applications. Amsterdam, Netherlands: Kluwer, 1999.
الجزء والصفحة:
...
4-3-2021
4309
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
![x^~=<span style=]() {Y_((N+1)/2) if N is odd; 1/2(Y_(N/2)+Y_(1+N/2)) if N is even " src="https://mathworld.wolfram.com/images/equations/OrderStatistic/NumberedEquation5.gif" style="height:50px; width:210px" /> |
(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://mathworld.wolfram.com/images/equations/OrderStatistic/NumberedEquation6.gif) |
(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.
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