ORDER%20STATISTICS

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ORDER STATISTICS

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ORDER STATISTICS

• Let X1, X2,,Xn be a r.s. of size n from a
  distribution of continuous type having pdf f(x),
  altxltb. Let X(1) be the smallest of Xi, X(2) be
  the second smallest of Xi,, and X(n) be the
  largest of Xi.

• X(i) is the i-th order statistic.

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ORDER STATISTICS

• It is often useful to consider ordered random
  sample.
• Example suppose a r.s. of five light bulbs is
  tested and the failure times are observed as
  (5,11,4,100,17). These will actually be observed
  in the order of (4,5,11,17,100). Interest might
  be on the kth smallest ordered observation, e.g.
  stop the experiment after kth failure. We might
  also be interested in joint distributions of two
  or more order statistics or functions of them
  (e.g. rangemax min)

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ORDER STATISTICS

• If X1, X2,,Xn is a r.s. of size n from a
  population with continuous pdf f(x), then the
  joint pdf of the order statistics X(1),
  X(2),,X(n) is

Order statistics are not independent.
The joint pdf of ordered sample is not same as
the joint pdf of unordered sample.
Future reference For discrete distributions, we
need to take ties into account (two Xs being
equal). See, Casella and Berger, 1990, pg 231.
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Example

• Suppose that X1, X2, X3 is a r.s. from a
  population with pdf
• f(x)2x for 0ltxlt1
• Find the joint pdf of order statistics and the
  marginal pdf of the smallest order statistic.

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ORDER STATISTICS

• The Maximum Order Statistic X(n)

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ORDER STATISTICS

• The Minimum Order Statistic X(1)

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ORDER STATISTICS

• k-th Order Statistic

of possible orderings n!/(k?1)!1!(n ? k)!
P(Xltyk)
P(Xgtyk)
fX(yk)
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Example

• Same example but now using the previous formulas
  (without taking the integrals) Suppose that X1,
  X2, X3 is a r.s. from a population with pdf
• f(x)2x for 0ltxlt1
• Find the marginal pdf of the smallest order
  statistic.

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Example

• XUniform(0,1). A r.s. of size n is taken. Find
  the p.d.f. of kth order statistic.
• Solution Let Yk be the kth order statistic.

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ORDER STATISTICS

• Joint p.d.f. of k-th and j-th Order Statistic
  (for kltj)

k-1 items
j-k-1 items
n-j items
of possible orderings n!/(k?1)!1!(j-k-1)!1!(n
? j)!
1 item
1 item
yj-1
yj
yj1
P(Xltyk)
P(ykltXltyj)
P(Xgtyj)
fX(yk)
fX(yj)