12. slides

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M u l t i v a r i a t e D i s t r i b u t i o n s

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L i m i t T h e o r e m s 1 - 1

L i m i t T h e o r e m s

C e n t r a l L i m i t T h e o r e m d e s c r i b e s t h e ( a s y m p t o t i c ) b e h a v i o u r o f

s a m p l e m e a n

X

1

, X2

, . . . , Xn

, i . i . d w i t h X

i

(, )

n ( x ) N

p

(0 , ) f o r n .T h e C L T c a n b e e a s i l y a p p l i e d f o r t e s t i n g .

N o r m a l d i s t r i b u t i o n p l a y s a c e n t r a l r o l e i n s t a t i s t i c s .

M V A : H u m b o l d t U n i v e r s i t t z u B e r l i n

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Asymptotic Distribution, N=5

-3 -2 -1 0 1 2 3

1000 Random. Samples

0

0.1

0.2

0.3

0.4

EstimatedandNormalDensity

Asymptotic Distribution, N=35

-2 0 2

1000 Random. Samples

0

0.1

0.2

0.3

0.4

EstimatedandNormalDensity

T h e C L T f o r B e r n o u l l i d i s t r i b u t e d r a n d o m v a r i a b l e s . S a m p l e s i z e

n = 5 ( l e f t ) a n d n = 3 5 ( r i g h t ) .

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T h e C L T i n t h e t w o - d i m e n s i o n a l c a s e . S a m p l e s i z e n

=5 ( l e f t ) a n d

n = 8 5 ( r i g h t ) .

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a c o n s i s t e n t e s t i m a t o r o f : P .x i s a s y m p t o t i c a l l y n o r m a l :

n

1

2 ( x ) Np

( 0 ,p

)

C o n d e n c e i n t e r v a l f o r ( u n i v a r i a t e ) m e a n

X

i

N

(, 2 )

n

x

N

(0

,1

)


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D e n e u

1 /2 a s t h e 1 /2 q u a n t i l e o f t h e N (0 , 1 ) d i s t r i b u t i o n .T h e n w e g e t t h e f o l l o w i n g 1

c o n d e n c e i n t e r v a l :

C

1 = x n

u

1 /2 ,x

+ n

u

1 /2 P ( C

1 ) 1 f o r n .


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EDF and CDF, n=100

-2 -1 0 1 2

x

0

0.

5

1

edf(x),cdf(x)

T h e s t a n d a r d n o r m a l c d f a n d t h e e m p i r i c a l d i s t r i b u t i o n f u n c t i o n f o r

n=

1 0 0 .

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EDF and CDF, n=1000

-2 0 2

x

0

0.

5

1

edf(x),cdf(x)

T h e s t a n d a r d n o r m a l c d f a n d t h e e m p i r i c a l d i s t r i b u t i o n f u n c t i o n f o r

n=

1 0 0 0

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B o o t s t r a p c o n d e n c e i n t e r v a l s

E m p i r i c a l d i s t r i b u t i o n f u n c t i o n

e d f F

n

=n

1

n

i

=1 I ( x i x )X

i

F

X i F n

x

= m e a n o f b o o t s t r a p s a m p l e

s u p

u P

n(

x

x

)

< u

P

n(

x )

< u

a . s .

0

C o n s t r u c t i o n o f C o n d e n c e I n t e r v a l s p o s s i b l e ! T h e u n k n o w n

d i s t r i b u t i o n o f x c a n b e a p p r o x i m a t e d b y t h e k n o w n d i s t r i b u t i o n o f

x

.


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EDF and 2 bootstrap EDFs, n=100

-2 -1 0 1 2 3

x

0

0.5

1

edfs{1..

3}(x)

T h e c d f F

n

a n d t w o b o o t s t r a p c d f ` s F

n

.

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L i m i t T h e o r e m s 1 - 1 0

T r a n s f o r m a t i o n o f S t a t i s t i c s

I f

n ( t ) N

p

( 0 , ) a n d i f f = ( f1

, . . . , fq

) : p qa r e r e a l v a l u e d f u n c t i o n s w h i c h a r e d i e r e n t i a b l e a t p , t h e n f ( t ) i s a s y m p t o t i c a l l y n o r m a l w i t h m e a n f () a n d c o v a r i a n c e

, i . e . ,

n{

f(

t)

f()}

N

q

(0

, )f o r n

,

w h e r e

= f j

t

i

(t

)t

=

( p q ) m a t r i x o f a l l p a r t i a l d e r i v a t i v e s .M V A : H u m b o l d t U n i v e r s i t t z u B e r l i n

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L i m i t T h e o r e m s 1 - 1 1

S u p p o s e

p = 2 , {Xi

} ni

=1 (, ); = 00

, = 1 0 . 50

.5 1

.W e h a v e b y C L T f o r n

n ( x ) N ( 0 , ).

T h e d i s t r i b u t i o n o f

x

2

1

x

2

x

1

+ 3 x2

? T h i s m e a n s t o c o n s i d e r

f

= (f

1

,f

2

)

w i t h

f

1

(x

1

,x

2

) =x

2

1

x

2

,f

2

(x

1

,x

2

) =x

1

+3 x

2

,q

=2

.


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L i m i t T h e o r e m s 1 - 1 2

T h e n f () =(

0

0

)a n d

= ( d i j ), d i j = f jx

i

x

== 2 x 1 1

1 3

x

=0

= 0 11 3

.W e h a v e t h e c o v a r i a n c e

0 11 3

1 121

2

1

0 11 3

= 1 72 72

1 3

.

T h i s y i e l d s

n

x

2

1

x

2

x

1

+ 3 x2

N

2

0

0

,

1 7

2

72

1 3

.


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H e a v y - T a i l e d D i s t r i b u t i o n s 2 - 1

H e a v y - T a i l e d D i s t r i b u t i o n s


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H e a v y - T a i l e d D i s t r i b u t i o n s

6 4 2 0 2 4 60.0

0.1

0.2

0.3

0.4

X

Y

2f 1f 1f 2f

q

q

Gaussy

Cauchy

Distribution Comparison


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S t u d e n t ' s t - d i s t r i b u t i o n

L e t X b e a n o r m a l l y d i s t r i b u t e d r a n d o m v a r i a b l e w i t h m e a n a n d

v a r i a n c e 2

, a n d Y b e t h e r a n d o m v a r i a b l e s u c h t h a t Y

2 / 2h a s a

c h i - s q u a r e d i s t r i b u t i o n w i t h n d e g r e e s o f f r e e d o m . A s s u m e t h a t X

a n d Y a r e i n d e p e n d e n t , t h e n

t

d e f

=X

n

Y

i s d i s t r i b u t e d a s S t u d e n t ' s t w i t h n d e g r e e s o f f r e e d o m .


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S t u d e n t ' s t - d i s t r i b u t i o n

T h e t - d i s t r i b u t i o n h a s t h e f o l l o w i n g d e n s i t y f u n c t i o n

f

t

(x

;n

) =

n

+12

n n2

1 + x2

n n +1

2

w h e r e n i s t h e n u m b e r o f d e g r e e s o f f r e e d o m ,

12. slides

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