pdf of the rth order statistics
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Transcript pdf of the rth order statistics
اعداد :د .زكية الصيعري
كلية العلوم للبنات
2014 -1435
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Dr. Zakeia A.Alsaiary
Refrences: المراجع
1- First course in order
statistics
Arnold, Balakrishnan and Nagaraja
2- Order statistics
H. A. David
3- اإلحصاءات الترتيبية
مكتبة المتنبي-للدكتورة ثروت محمد عبدالمنعم
Dr. Zakeia
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Dr. Zakeia A.Alsaiary
Definition
The order statistics of a random sample
X1, . . . ,Xn are the sample values placed in
ascending order. They are denoted by X(1), . .
. ,X(n).
The smallest of the Xi,s is denoted by X1:n,
the second smallest is denoted by X2:n,…,
and finally the largest is denoted by Xn:n thus
order statistics are random variables that
satisfy X(1) < X(2) <· · · · < X(n). The
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اإلحصاءات المرتبة :هي عناصر عينة عشوائية مرتبة من األصغر إلى
األكبر .وفي أغلب مناقشاتنا لإلحصاءات المرتبة سوف نعتبر العينة
العشوائية تتبع توزيعات متصلة.
Example:
Let x1 = 0.62 , x2 = 0.98, x3= 0.31 , x4 = 0.81 ,
x5 = 0.53 are observation for independent
expeirement , the order statistics of it are:
=Y1 = 0.31 , y2 = 0.53 , y3 = 0.62 , y4 = 0.81 , y5
0.98
Y3 = 0.62 is the median and
the range = y5 - y1= 0.98 – 0.31 = 0.67
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ORDER STATISTICS
• If X1, X2,…,Xn be 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
g y 1 , y 2 ,
, y n n !f y 1 f y 2
f y n
for y (1) ... y ( n )
=0
g y 1 , y 2 ,
(elsewhere)
, y n n ! f y i
n
i 1
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ORDER STATISTICS
Example 1:4 كتاب اإلحصاءات الترتيبية صفحة
Find the joint pdf of the order statistics for the
uniform distribution , the standard exponential
distribution and normal distribution?
Solution: p.d.f for the uniform is:
f ( x ) 1, 0 x 1
g ( y 1 , y 1,..., y n ) n !, 0 y 1 y 1 ..., y n 1,
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ORDER STATISTICS
Solution: p.d.f for the standard Exponential
distribution is:
f (x ) e
g ( y 1 , y 1 ,..., y n ) n !e
x
, x 0
n
yi
i 1
, 0 y 1 y 1 ..., y n ,
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ORDER STATISTICS
Solution: p.d.f for the standard normal distribution
is:
x2
f (x )
1
e
2
2
, x
n
y i2
2
i 1
n!
g ( y 1 , y 1 ,..., y n )
e
, y 1 y 1 ..., y n ,
2
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The marginal distributions for
the Oder Statistics
p.d.f of the rth order statistics:
Theorem:
If X1, X2,…,Xn be a r.s. of size n from a population
with continuous pdf f(x), then the p.d.f. of the rth
order statistics X(r) is given as:
n!
gr ( y r )
f (yr)
( r 1)!( n r )!
r 1
n 1
[ F ( y r ) ] [1 F ( y r ) ] , y r
PROOF:
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ORDER STATISTICS
• r-th Order Statistic
y1 y2 … yr-1 yr yr+1 …
P(X<yr)
y
yn
P(X>yr)
# of possible orderings
n!/{(r1)!1!(n r)!}
f(yr)
gr ( y r )
n!
f (yr)
( r 1)!( n r )!
[ F ( y r ) ]r 1 [1 F ( y r ) ]n 1 , y r
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The marginal distributions for the oder
statistics
p.d.f of the largest order statistics:
Theorem:
If X1, X2,…,Xn be a r.s. of size n from a population
with continuous pdf f(x), then the p.d.f. of the
Largest order statistics Y(n) is given as:
n 1
g n ( y n ) n f ( y n )[ F ( y n )] , y n
PROOF:
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The marginal distributions for
the oder statistics
p.d.f of the smallest order statistics:
Theorem:
If X1, X2,…,Xn be a r.s. of size n from a population
with continuous pdf f(x), then the p.d.f. of the
smallest order statistics X(1) is given as:
n 1
g1 ( y 1 ) n f ( y 1 )[1 F ( y 1 )] , y n
PROOF:
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Example
Let
Y 1 Y 2 ... Y 6
are an O. S. of sample size n = 6
and the p.d.f. of this sample is
1
f (x )
, 0 x 2
2
Find: g r ( y r ) , g1 ( y 1 ) , g 6 ( y 6 )
Solution:
6!
r 1
61
gr ( y r )
y r [2 y r ] , 0 y r 2
6
( r 1)!( 6 r )!2
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6
5
g1 ( y 1 ) 6 [ 2 y 1 ] , 0 y 1 2
2
6 5
g6 ( y 6 ) 6 y 6 , 0 y 6 2
2
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Example
Let X
,X 2 ,X 3 ,X 4 ,X 5
is a random sample from beta
population with parameters 1 , 2 .
Y 1 Y 2 Y 3 Y 4 Y 5 Let the order statistics of the
sample,
1
f ( x ) 2 x , 0 x 1
Find:
g n ( y n ) , g 1 ( y 1 ) then find P (Y 0.2 ),
g 3 ( y 3 ) then
find the var iance
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Solution:
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Solution
f (x )
1
x 1 (1 x ) 1 , 0 x 1 , 1, 0
( , )
f ( x ) 2 x , 0 x 1
0 , x 0
2
F ( x ) x , 0 x 1
1 , x 1
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Example
Let
is a random sample from
standard uniform distribution.
Find: p.d.f. for the median?
________________________________________
Y m 1
m
m
X 1 , X 2 ,..., X 2m 1
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Joint p.d.f. of i-th and j-th Order
Statistic (for i<j)
Theorem:
If X1, X2,…,Xn be a r.s. of size n from a population
with continuous pdf f(x), and Y1< Y2<…<Yn are
the order statistics of that sample, then the p.d.f.
of the two order statistics Yi< Yj , i<j and i,j = 1,2,
…,n is given as
g ij ( y i , y j )
n!
[F ( y i )]i 1 f ( y i )[F ( y j ) F ( y i )] j i 1 f ( y j )[1 F ( y j )]n j
(i 1)!( j i 1)!(n j )!
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ORDER STATISTICS
• Joint p.d.f. of i-th and j-th Order Statistic (for i<j)
i-1 items
y1 y2 … yi-1
1 item
n-j items
1 item
# of possible orderings
n!/{(i1)!1!(j-i-1)!1!(n j)!}
y
yi
P(X<yi)
yi+1
…
yj-1 yj yj+1
P(yi<X<yj)
f(yi)
g ij ( y i , y j )
j-i-1 items
yn
P(X>yj)
f(yj)
n!
[F ( y i )]i 1 f ( y i )[F ( y j ) F ( y i )] j i 1 f ( y j )[1 F ( y j )]n j
(i 1)!( j i 1)!(n j )!
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Example
Let
is a random sample from beta
population with parameters 1 , 2 .
Let Y 1 Y 2 Y 3 Y 4 Y 5 the order statistics of the sample,
X 1 ,X 2 ,X 3 ,X 4 ,X 5
f ( x ) 2 x , 0 x 1
Find: 1- Joint p.d.f. for y 2 , y 4 ( g 2,4 ( y 2 , y 4 ) )
1
1
2
2- Let n = 4 then find P ( y 3 ) , P ( y 3 )
2
3
3
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Example
Let Y
Y 2 Y 3 Y 4 Y 5
the order statistics of random
sample with size n = 5, from distribution with
p.d.f.
1
f ( x ) e x , 0 x
Prove that the two O.S. Z 2 Y 4 Y 2 , Z 1 Y 2
Are independent.
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Solution
f ( x ) e x , 0 x
0 , x 0
F ( x ) 1 e x , 0 x 1
1 , x 1
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g ( y 2 , y 4 ) 120e y 2e 2 y 4 (1e y 2 )(e y 2 e y 4 ) , 0 y 2 y 4
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c.d.f of the rth order statistics:
Theorem:
If X1, X2,…,Xn be an independent and identical r.s.
of size n from a population with pdf f(x) and cdf
F(x), then the c.d.f. of the rth order statistics X(r)
is given as:
r j
n r n r
n!
[
F
(
x
)]
j
Fr :n ( x )
(
1)
j
( r 1)!( n r )! j 0
rj
PROOF:
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p.d.f of the rth order statistics:
Special cases:
1 Fr :n ( ) 0
n r n r
n!
1
j
2 Fr :n ( )
( 1)
( r 1)!( n r )! j 0 j
rj
1 , prove that ??
3 Fn :n ( x ) [ F ( x ) ]n why ??
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c.d.f of the rth order statistics:
Theorem:
If X1, X2,…,Xn be an independent and identical r.s.
of size n from a population with pdf f(x) and cdf
F(x), then the c.d.f. of the rth order statistics X(r)
is given as:
n r j 1
r 1 r 1
n!
1
[1
F
(
x
)
]
j
Fr :n ( x )
(
1)
j
( r 1)!( n r )! j 0
n r j 1
PROOF:
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p.d.f of the rth order statistics:
Special case:
F1:n ( x ) 1 [1 F ( x )]
n
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why ??
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p.d.f of the rth order
statistics:
صور أخرى لدالة التوزيع التراكمية لإلحصاء المرتب الرائي
:مثبتة في الكتب
1
Fr :n ( x )
( r , n r 1)
F (x )
y r 1 [1 y ]n r dy
0
I F ( x ) ( r , n r 1)
n
j
nj
Fr :n ( x ) [ F ( x ) ] [1 F ( x ) ]
j
j
n
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Example
Let Y 1 Y 2 Y 3 Y 4 Y 5 the order statistics of the sample
with n = 5, from the distribution
f ( x ) 2 x , 0 x 1
1
Find: P ( y 4 )
2
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Example
Let Y 1 Y 2 Y 3 Y 4 Y 5 Y 6 Y 7 the order statistics of the
sample with n = 5, from the distribution
f ( x ) 3(1 x )2 , 0 x 1
Find: P ( y 4 1 3 0.6)
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