Kolmogorov goodness-of-fit test ! for -symmetric distributions - - PowerPoint PPT Presentation

Kolmogorov goodness-of-fit test for -symmetric distributions in climate and weather modeling

Reporter: Dr. Zhanna Zenkova PhD, MBA, TSU Associate Professor

Authors: Z.N. Zenkova L.A. Lanshakova

S !

Agenda

Title: Kolmogorov goodness-of-fit test for -symmetric distributions in climate and weather modeling

• 1. Mathematical statistics in climate and weather

modeling;

• 2. Kolmogorov goodness-of-fit test;
• 4. Example.

S !

• 3. -symmetric distributions;

S !

3

Mathematical statistics in climate and weather modeling

Water lifting

4

Mathematical statistics in climate and weather modeling

Уровень подъема воды

Kolmogorov goodness-of-fit test

5

( )

N

X X X X ,..., ,

2 1

=

) ( ) ( sup x G x F d

N x N = R

,

) ( ) ( : x G x F H =

( )(

)

=

=

N i i x N

X I N x F

1 ;

1 ) (

(1) (2)

( )

( )

+ =

= = <

k z k k N N

e z K z d N

2 2

2

) 1 ( P lim

(3)

Kolmogorov goodness-of-fit test

6

) (x FN ) (x G

N

d

• symmetric distribution

7

,

(4)

S !

• Definition. If for

, , k >1, for and c.d.f. F(x) satisfies the conditions: where are continuous monotonically decreasing functions, , , , then c.d.f. F(x) is -symmetric c.d.f.

k

c c c < < < ...

2 1

k i , 1 =

) ( ) ( + = =

i i i

c F c F p

, ) ( 0 = = c F p

, 1 ) (

1 1

= =

+ + k k

c F p k j , 1 =

1 + j

c x

{ } ( ) { } ( ),

) ( , min ) ( , max

1 1

+ = +

+ +

x S x F p p p p x S x F

j j j j j j

) (x S j

( )

) ( ) (

1

x S x S

j j

=

1 + j

c x

( )

i i j

c c S =

( )

1

c c S

j j

=

+

• symmetric distribution

8

(5)

S !

For k = 1, , we can obtain classical symmetry of c.d.f. around :

5 . ) ( 1

1

= = c F p

x c x S =

1 1

2 ) (

1

c

( )

x c F x F =

1

2 1 ) (

Test modification

9

against

:

S

H

), ( ) ( x G x F =

, ,

S

G F

R x

:

1 S

H

, , ,

S

G F V

R x

( ),

) ( ) ( x G V x F =

(6)

=

S N

d

) ( ) ( sup

* 1

x G x F p

S N x R

1

) ( *

) (

p x G

x G =

(7)

Test modification

10

(8) is e.d.f. based on k times symmetrized sample

) (x F S

N

( )

* *,...,

1 N

X X

=

*

X

( ) { }

1 1 ) 1 ( ) (

, min

+

=

j i j k j i j i

X S X X

,

) ( i i

X X =

) ( * k i

i

X X =

N i , 1 =

, , 1 k j =

Test modification

11

(9) For

( ),

) ( ) ( x G V x F =

, > y

( )

y d P

S N <

( )

( )

, , 1 1 1 1 1 1

1 det !

N i,j i j j i

) i j p a p V p b p V N

= +

+ = ! ( 1 1

,

1 1

= p y N i ai 1

, 1 1

1 1

+ = p z N i bi

N i , 1 =

, , max

1 1 1

=

+ + + j j j j j j

z p p p y y

, , min

1 1 1

=

+ + + j j j j j j

y p p p z z

k j , 1 =

y z y

k k

= =

+ + 1 1

Test modification

12

(10) For

> z

( )

, P lim

1

= < p z K z d N

S N N

Example

13

(11) The uniform c.d.f. U(x) is satisfied by (4), if for arbitrary

1 1

p c =

, 1

1 <

< p

< = . 1 , 1 1 , , 1 1 ) (

1 1 1 1 1 1 1

x c p p x c x x p p x S

Example

14

against S

H0

[ ],

1 , , ) ( ) ( : = = x x x U x F

[ ],

1 , ), ( ) ( :

1

= x x U x F

S

H1

(12)

[ ]

( ) (

]

= . 1 , , 1 1 1 1 , , , ) (

1 1 1 1 1 1 1

p x p x p p x p x p x U

m m

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

Example

15

) (x U

) (

1 x

U

8 .

1 1

= = p c

m = 3,

Classical symmetric cdf,

5 .

1 1

= = p c

Example

16

m = 3,

7 ,

1 =

p

Thank you for your attention!