Slides for Chapter 2

Order and Rank Statistics
Chapter 2
Order Statistics
• Assume that a sample of size 𝑛 is selected
from the population.
𝑋1 , 𝑋2 , ⋯ , 𝑋𝑛
• The 𝑘th order statistic of a statistical sample is
equal to its 𝑘th-smallest value.
Parametrik Olmayan Yöntemler 4. Baskı
Gamgam, H. & Altunkaynak B
Edited by Güler, H.
2
Order Statistics
• If 𝑋 1 is the smallest value in the sample, 𝑋 2
is the second smallest value, and so on, then
the order statistics are
𝑋1 <𝑋2 <⋯<𝑋𝑛.
• 𝑋 𝑘 : 𝑘th-smallest value in the sample is the
𝑘th order statistic of this sample.
Parametrik Olmayan Yöntemler 4. Baskı
Gamgam, H. & Altunkaynak B
Edited by Güler, H.
3
Example
• Assume that a sample of 5 observations is
9, –3, 4, 1, 0
• Then 𝑥1 = 9, 𝑥2 = −3, 𝑥3 = 4, 𝑥4 = 1, 𝑥5 = 0.
• The ordered sample is
–3, 0, 1, 4, 9
• Therefore, the order statistics are
𝑥 1 = −3, 𝑥 2 = 0, 𝑥 3 = 1, 𝑥 4 = 4, 𝑥 5 = 9.
Parametrik Olmayan Yöntemler 4. Baskı
Gamgam, H. & Altunkaynak B
Edited by Güler, H.
4
Order Statistics and Their Usage
• Together with rank statistics, order statistics are
among the most fundamental tools in nonparametric statistics and inference.
• They are also used to compute some important
statistics:
• The first order statistic is always the minimum
of the sample, that is,
𝑋 1 = 𝑚𝑖𝑛 𝑋1 , 𝑋2 , ⋯ , 𝑋𝑛 .
Parametrik Olmayan Yöntemler 4. Baskı
Gamgam, H. & Altunkaynak B
Edited by Güler, H.
5
Order Statistics and Their Usage
• Similarly, for a sample of size 𝑛, the 𝑛th order
statistic is the maximum, that is,
𝑋 𝑛 = 𝑚𝑎𝑥 𝑋1 , 𝑋2 , ⋯ , 𝑋𝑛 .
• The range of a sample of size 𝑛, is the difference
between the 𝑛th and 1st order statistics, that is,
𝑅 =𝑋 𝑛 −𝑋 1 .
Parametrik Olmayan Yöntemler 4. Baskı
Gamgam, H. & Altunkaynak B
Edited by Güler, H.
6
Order Statistics and Their Usage
• The median of a sample is defined in terms of
order statistics:
𝑋 𝑛+1 ,
if 𝑛 is odd
2
𝑀= 𝑋
𝑛
2
+𝑋
2
𝑛
+1
2
,
Parametrik Olmayan Yöntemler 4. Baskı
Gamgam, H. & Altunkaynak B
Edited by Güler, H.
if 𝑛 is even
7
Order Statistics and Their Usage
• The joint (and marginal) distribution of the
order statistics is different from the joint (and
marginal) distribution of the population.
• Order statistics are dependent.
Parametrik Olmayan Yöntemler 4. Baskı
Gamgam, H. & Altunkaynak B
Edited by Güler, H.
8
Example
• Following is a sample of midterm scores:
𝒊 𝑿𝒊
1 60
𝒊
𝑿
𝒊
2 55
3 80
4 45
1
2
3
4
45
55
60
65
5 70
6 65
7 70
5
6
7
70
70
80
𝑋
1
= 45
𝑋
4
= 65
𝑋
7
= 80
Parametrik Olmayan Yöntemler 4. Baskı
Gamgam, H. & Altunkaynak B
Edited by Güler, H.
𝑅 =𝑋 𝑛 −𝑋 1
= 80 − 45 = 35
9
Rank Order Statistics
• If 𝑋1 , 𝑋2 , ⋯ , 𝑋𝑛 is a sample of size 𝑛, then the
rank order statistics are represented by
𝑟 𝑋1 , 𝑟 𝑋2 , ⋯ , 𝑟 𝑋𝑛 .
• In general, if 𝑥𝑖 − 𝑥𝑗 = 𝑢 and
1 ,𝑢 ≥ 0
𝑠 𝑢 =
0 ,𝑢 < 0
then
𝑟 𝑋𝑖 = 𝑛𝑗=1 𝑠 𝑥𝑖 − 𝑥𝑗 = 1 + 𝑛𝑖≠𝑗 𝑠 𝑥𝑖 − 𝑥𝑗 .
Parametrik Olmayan Yöntemler 4. Baskı
Gamgam, H. & Altunkaynak B
Edited by Güler, H.
10
Example
• Assume the following sample is observed:
𝑥1 = 30, 𝑥2 = 40, 𝑥3 = 5, 𝑥4 = 80, 𝑥5 = 51, 𝑥6 = 35.
𝑟 𝑋1 =
𝑟 𝑋2 =
𝑟 𝑋3 =
𝑟 𝑋4 =
𝑟 𝑋5 =
𝑟 𝑋6 =
𝑛
𝑗=1 𝑠
𝑛
𝑗=1 𝑠
𝑛
𝑗=1 𝑠
𝑛
𝑗=1 𝑠
𝑛
𝑗=1 𝑠
𝑛
𝑗=1 𝑠
𝑥1 − 𝑥𝑗 = 1 + 0 + 1 + 0 + 0 + 0 = 2.
𝑥2 − 𝑥𝑗 = 1 + 1 + 1 + 0 + 0 + 1 = 4.
𝑥3 − 𝑥𝑗 = 0 + 0 + 1 + 0 + 0 + 0 = 1.
𝑥4 − 𝑥𝑗 = 1 + 1 + 1 + 1 + 1 + 1 = 6.
𝑥5 − 𝑥𝑗 = 1 + 1 + 1 + 0 + 1 + 1 = 5.
𝑥6 − 𝑥𝑗 = 1 + 0 + 1 + 0 + 0 + 1 = 3.
Parametrik Olmayan Yöntemler 4. Baskı
Gamgam, H. & Altunkaynak B
Edited by Güler, H.
11
Example
𝑥1 = 30, 𝑥2 = 40, 𝑥3 = 5,
𝑥4 = 80, 𝑥5 = 51, 𝑥6 = 35
𝑥 1 = 5, 𝑥 2 = 30, 𝑥 3 = 35,
𝑥 4 = 40, 𝑥 5 = 51, 𝑥 6 = 80
𝑟 𝑋1 = 2, 𝑟 𝑋2 = 4, 𝑟 𝑋3 = 1,
𝑟 𝑋4 = 6, 𝑟 𝑋5 = 5, 𝑟 𝑋6 = 3.
Parametrik Olmayan Yöntemler 4. Baskı
Gamgam, H. & Altunkaynak B
Edited by Güler, H.
12

Download Report