Zenkova Zh.N., L.A. Lanshakova
National Research Tomsk State University, Tomsk , Russia E-mail: thank off@fpmk .tsu.ru
Introduction
Climate and weather modeling is a complex task which involves a wide range of different methods and techniques (Stocker, 2011; McGuffie, Henderson-Sellers, 2005). The most important of them is a statistical data treatment (Hennemuth et al., 2013; von Storch, Zwiers, 1999; Haltiner, Williams 1980). It allows describing, analyzing and forecasting weather processes, helping to detect and scientifically prove implicit, subtle climate changes.
Despite the plenty of observations and collected information in the field of weather and climate modeling, there are many problems dealing with inaccuracy of resulted issues (Lupo, Market, 2002; Martner, Politovich, 1999; Zhu, Halton, 2014), because statistically it is impossible to find exact values. So there are many researches devoted to improvement of statistical procedures. In (Zenkova, 2015) it is given a short review of methods which allows to increase an accuracy of statistical treatment using additional and auxiliary information, especially for incomplete and censored data (Klein, Moeschberger, 2010).
In this paper authors considered a new definition of
S r
-symmetry of cumulative distribution function (c.d.f.) and used it in modification of Kolmogorov goodness-of-fit test. The distributions of modified statistic were given for null and alternative hypotheses for asymptotical and non- asymptotical cases. An example of power increasing is given.
Kolmogorov goodness-of-fit test for
S r
-symmetric distribution
The classical test (Kolmogorov, 1933) solves weather the unknown cumulative distribution function (c.d.f.) F(x) of a sample
X = ( X
1, X
2,..., X
N)
is equal to a hypothetical c.d.f. G(x). It is based on statisticsd
N=sup F
N( x ) G ( x )
x
∈R
−
, where
F
N(x )
is empirical distribution function (e.d.f.) (D’Agostino, Stephens, 1986). The asymptotical null distribution ofd
N can be calculated for z >0 by means of the formula:( ) ( ) ( 1 ) .
P
lim
+∞∑
2 2 2−∞
=
−
∞
→
< = = −
k
z k k
N
N d
Nz K z e
(1)In this paper the authors considered the hypothesis:
H
0S:F ( x ) = G ( x )
,F , G ∈ Ξ
S,∀ x ∈ R
, againstH
1S:F ( x ) = V ( G ( x ) )
,F , V , G ∈ Ξ
S,∀ x ∈ R
, whereΞ
S is a set ofS r
-symmetric c.d.f.
Definition. If for
c
1< c
2< ... < c
k,i = 1 , k
, k >1,p
i= F ( c
i) = F ( c
i+ 0 )
,p
0= F ( c
0) = 0 , ,
1 ) (
11
=
+=
+ k
k
F c
p
forj = 1 , k
andx ≤ c
j+1 c.d.f. F(x) satisfies the conditions:{ }
( max , ( ) 0 )
1 1− ( min { , ( ) + 0 } ) ,
−
=
+
+ +F x S x
p p p p
x S x
F
jj j j
j
j (2)
where
S
j(x )
are continuous monotonically decreasing functions,( ) S
j −1( x ) = S
j( x )
,x ≤ c
j+1,k
j = 1 ,
,S
j( ) c
i= c
i,i = 1 , k
,S
j( ) c
j+1= c
0, then c.d.f. F(x) isS r
-symmetric c.d.f.
For k = 1,
p
1= F ( c
1) = 0 . 5
,S
1( x ) = 2 c
1− x
we can obtain classical symmetry of c.d.f. aroundc
1:( c x )
F x
F ( ) = 1 − 2
1−
.For testing the hypothesis
H
0S it can be used the modified Kolmogorov statisticS
d
N=
p
1sup F
NS( x ) G
*( x )
x
∈R
−
, (3)
where
1 )
* (
)
( x
G x pG =
,F
NS(x )
is e.d.f. based on k times symmetrizied sampleX
*=( X
1*,..., X
*N)
.Here for
i = 1 , N
i* (k)X
iX =
,X
i(j)= min { X
i(j−1), S
k−j+1( ) X
ij−1}
,j = 1 , k
,X
i(0)= X
i. It is easy to prove that random variableX
i* has c.d.f. *( ) min {
( )p1, 1 }
x
x
FF =
.The distribution of
d
NS can be calculated forF ( x ) = V ( G ( x ) ) , y > 0
, by the following formula:( d y )
P
NS<
=N!det( ) ( )
, , 1 1
1 1 1
1
1
N i,j i j i j
) i j
p a p V p
b p V
= +
−
+
−
∧
−
∧
∨
! (
1 1
0
(4)
where
∨ −
∧
=
1 1
p y N
a
i1 0 i
,
∧ − +
∨
=
1
1
11
0 p
z N
b
ii
,i = 1 , N
,
= −
++ + 1
1 1
,
max
jj j
j j
j
z
p p
y p
y
,
= −
++ + 1
1 1
,
min
jj j
j j
j
y
p p
z p
z
,j = 1 , k
,y
k+1= z
k+1= y
.If in (4) V(t) = t then it allows to find the null distribution of
d
NS. The asymptotical null distribution of modified statistic (3) can be calculated for z >0 as( ) ,
P lim
1
=
∞
<
→
p
K z z d N
NSN here
K (z )
is as in (1).Example
The uniform c.d.f. U(x) is satisfied (2) if for arbitrary
c
1= p
1,0 < p
1< 1
,
≤
<
− ⋅
−
≤
≤
− ⋅
−
=
. 1 1 ,
1
, 0
1 , 1 ) (
1 1 1
1 1
1
1
x c p p
x
c x p x
p x
S
For
x ∈ [0,1]
let consider a hypothesisH
0S:F ( x ) = U ( x ) = x
againstH
1S:F ( x ) = U
1( x ),
wherefor integer m >0
)
1
( x U
=[ ]
( ) ( ]
∈
−
− −
−
∈
. 1 , 1 ,
1 1 1
, , 0 ,
1 1
1
1 1
1
p p x
p x
p p x
p x
m m
Function
U
1( x )
is also satisfied (2) for the sameS
1( x )
. The plots of functionsU (x )
andU
1( x )
for8
.
1
0
1
= p =
c
are demonstrated on fig. 1.Fig.1.
U (x )
andU
1( x )
forc
1= p
1= 0 . 8
The dependences of powers of modified (Ws) and non-modified (W) tests from a sample size N for p- value = 0.05, m = 3,
p
1 = 0.7 are shown on fig. 2. It is obvious that modified test is more powerful than classical one.Fig. 2. The powers of Kolmogorov goodness-of-fit tests: modified (Ws) and non-modified (W) for p-value = 0.05, m = 3,
p
1 = 0.7Conclusions
In the paper the new definition of
S r
-symmetric cumulative distribution function was given, it was con- sidered the way of involving the additional information about
S r
-symmetry to modify Kolmogorov goodness-of- fit test. The precise null and alternative distributions and asymptotical null distribution of the modified statistic are given. It was demonstrated the example which prove that the modification of the statistic helps to increase the power of the new test.
Therefore, using additional information about
S r
-symmetry allows to construct more effective and pow- erful statistical procedures that also can be very profitable in climate and weather modeling, because it can help to find more accurate and precise results of data treatment, obtain better forecasting and increase certainties of issues.
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