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order statistics∗
CWoo†
2013-03-21 17:47:09
Let X1 , . . . , Xn be random variables with realizations in R. Given an outcome ω, order xi = Xi (ω) in non-decreasing order so that
x(1) ≤ x(2) ≤ · · · ≤ x(n) .
Note that x(1) = min(x1 , . . . , xn ) and x(n) = max(x1 , . . . , xn ). Then each
X(i) , such that X(i) (ω) = x(i) , is a random variable. Statistics defined by
X(1) , . . . , X(n) are called order statistics of X1 , . . . , Xn . If all the orderings are
strict, then X(1) , . . . , X(n) are the order statistics of X1 , . . . , Xn . Furthermore,
each X(i) is called the ith order statistic of X1 , . . . , Xn .
Remark. If X1 , . . . , Xn are iid as X with probability density function fX
(assuming X is a continuous random variable), Let Z be the vector of the order
statistics (X(1) , . . . , X(n) ) (with strict orderings), then one can show that the
joint probability density function fZ of the order statistics is:
fZ (z) = n!
n
Y
fX (zi ),
i=1
where z = (z1 , . . . , zn ) and z1 < · · · < zn .
∗ hOrderStatisticsi
created: h2013-03-21i by: hCWooi version: h36111i Privacy setting:
h1i hDefinitioni h62G30i
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