Chapter 4
Some Elementary Statistical Inference
4.4 Order Statistics
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Boxiang Wang
Chapter 4 STAT 4101 Spring 2021
Outline
1 Motivations and definitions. Slide 3 – Slide 4
2 Joint pdf of order statistics. Slide 5 – Slide 12
3 Applications: (Slide 13 – Slide 28) Quantiles.
Graphical applications on q-q plot and boxplot. Confidence interval for medians.
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Boxiang Wang
Chapter 4 STAT 4101 Spring 2021
Definitions of order statistics
Suppose that X1, · · · , Xn is a random sample.
Order statistics: Y1, Y2, . . . , Yn: the kth order statistic Yk of this
random sample is equal to the k-th smallest value.
Most commonly used order statistics are : Yn = max(X1, . . . , Xn),
Y1 = min(X1,…,Xn).
More examples:
1 range of the random sample: Yn − Y1.
2 midrange of the random sample: Yn+Y1 . 2
3 median: Y(n+1)/2 when n is odd; Yn/2+Y(n/2+1) 2
even.
when n is
In some other occasions, you may see the notation of order statistics as X(1), X(2), . . . , X(n).
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Boxiang Wang
Chapter 4 STAT 4101 Spring 2021
A motivating example
Let X1, X2, X3 be iid with common pdf f(x) = exp(−x), 0 < x < ∞, zero elsewhere. Find the distribution of
Y =min(X1,X2,X3).
Solution:
We have
P (min(X1, X2, X3) ≤ y) = 1 − P (min(X1, X2, X3) > y)
= 1−P(X1 > y)P(X2 > y)P(X3 > y)
= 1 − e−3y.
Hence Y follows an exponential distribution with the expectation
1/3.
Recall f(x) = βe−βx and F(x) = 1 − e−βx.
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Boxiang Wang
Chapter 4 STAT 4101 Spring 2021
Theorem
Let Y1, · · · , Yn denote the n order statistics based on the random sample X1,··· ,Xn from a continuous distribution with pdf f(x) and support (a, b). For k ̸= 1, n, the density of Yk is given by
gYk(yk) = n! [F(yk)]k−1[1−F(yk)]n−kf(yk) Ia
p3 =P(yk
One-to-one transformation: x1 = y1y2 and x2 = y1(1 − y2), 0