The Consistent Estimators for Homogeneous Gaussian Fields Statistical Structures

IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue 3 Ver. 1 (May. - June. 2017), PP 01-06 www.iosrjournals.org

The Consistent Estimators for Homogeneous Gaussian Fields Statistical Structures Z. Zerakidze, M. Patsasia.

Gori University, Gori, Georgia, Sukhumi University, Tbilisi, Georgia.

Abstract: The present theory of consistent estimators of the parameters of statistical structures of homogeneous Gaussian fields can be used, for example, in the reliability predication of different engineering designs. In the paper there are discussed Gaussian homogeneous fields statistical structures E, S , i , i  I  in Banach space of measures. We prove necessary and sufficient conditions for existence of such estimators.

Keywords: consistent estimators, orthogonal, weakly separable, strongly separable statistical structures.

Classification 62HO5, 62H12

I. Introduction

Let there is given

E, S  measurable space and on this space there given i , i  I family of probability

measures depended on S, the I set of parameters.

Let bring some definition (see [1]-[10]).

Definition 1. A statistical structure is called object E, S , i , i  I  where i  I some parameter associated

i , I set of parameters.

Definition 2. A statistical structure E, S , i , i  I  is called orthogonal (singular) if i and  j are orthogonal for each i  j, i  I , j  I .

Definition 3. A statistical structure E, S , i , i  I  is called weakly separable if there exists family Smeasurable sets X i , i  I  such that relations are fulfilled: 1, if i  j i j i  I & j  I   i X j    . 0, if i  j Definition4. A statistical structure E, S , i , i  I  is called strongly separable if there exists disjoint family Smeasurable sets X i , i  I  such that the relation are fulfilled.  i  X i   1, i  I . with probability measure

Remark 1. A strong separable there follows weakly separable. From weakly separable there follows orthogonal but not vice versa (see [1]-[4]).

Remark 2. In the general theory of statistical decisions there often arises a problem of transition from a weakly separated family of probability measures to the corresponding strongly separated family. In 1981, A. Skozokhod (see [1]) proved (in ZFC & CH theory) proved that if the Continuum Hypothesis is true, then an arbitrary weakly separated family of probability measures,whose cardinality is not greater than the cardinality of the continuum is strongly separable.The validity of the inverse relation was established in (see [9]-[10]). In particular, It was shown there that if an arbitrary weakly separated family of probability measures whose cardinality is less than or equal to the cardinality of the continuum is strongly separated, then the continuum Hypothesis ( CH ) is true Applying Martin’s axiom (MA) in 1984 Z. Zerekidze(see [2]-[4]) proved that an arbitrary weakly separated family of Borel probability measures defined in a separable completely metrizable space ( i. e. Polish space) is strongly separated if its cordinality is not prated than the cardinality of the continuum. G. Pantsulaia proved (see [9]-[10]) this result is extended to all complete metric spaces whose topological weights are not measurable in a wider sence. Z. Zerekidze proved in ZF theory (see [8]) for the statistical structure E, S , i , i  I  where N set at natural numbers, orthogonality, weak separability, separability and strong separability equivalent concepts.

Let I be set of parameters and B (I)  -algebra of subsets of I which contains all finite subsets I.

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The Consistent Estimators For Homogeneous Gaussian Fields Statistical Structures Definite 5. A statistical structure E, S , i , i  I  will be said to admit a consistent estimators of parameter

iI

if

there

exists

at

least

one

measurable

i x :  x   i  1, i  I .

We denote by i the completion of the measure i

 : E, S   I , BI 

map

and dom i the

 -algebra of all i

such

that

measureable subsets

of E Remark 3. By A. Skorokhod was introduced definition a consistent estimators of parameters (see [1]).

Definite 6. A statistical structure E, S , i , i  I  will be said to admit a consistent estimators of any

parametric function if for any real bounded measurable function g : I , BI   R there exists at least one

i x : f x   g i   1, i  I . .

measurable function f : E , S   R such that Definition 7. A statistical structure

E, S , i , i  I  will

be said to admit an unbiased estimators of any

parametric function if for any real bounded measurable function g : I , BI   R there exists at least one measurable function

 : E, S   R, such that   x i dx   g i , i  I .

B

Remark 4. If a statistical structure E, S , i , i  I  admitting a consistent estimators of parameters i  I then this statistical structure E, S , i , i  I  which admits a consistent estimators for any parametric function and a statistical structure which admits an unbiased estimators of any parametric function (see [6]-[8]).

Let

M

be a real linear space of all alternating finite measures on S .

Definition 8.A linear subset

M   M  is called a Banach space of measures if:

1) A norm can be defined on M  so that M  will be a Banach space with respect to this norm,and for

 ,  M  , and real number   0 the enequality      is

any orthogonal measures fulfilled; 2) If

  M B , f x   1 than  f  A   f x  dx   M B and  f  

;

A

 n  M B ,  n  0,  n E   , n  1,2,...

3) If

and

 n  0 then for any linear functional

  M lim   n   0 , where M conjugate to M B linear space. 

 B

 B



n 

The definition and construction of the Banach space of measures is studied Z.Zerekidze(see [7]).

M Bi Banach space i  I .

Definition 9. Let I some set of indexes and



set M B  X i iI , X i  M Bi ,

 Then the M B with norm

 iI

X i iI

Banach spaces M Bi and denoted so

Xi

M Bi

 iI

We

   .  Xi   is the Banach space. It is called the direct sum of M Bi

M B   M Bi . iI

By Z. Zerakidze was introduced definition and is studied construction the Banach space of measures (see [7]) The following theorem has also been proved in this paper (see [7]).

Theorem 1. Let M B be a Banach space of measures, then in M B there exists a family of pairwise orthogonal probability measures M  i , i  I  such that



of the norm :  B  

M B   M Bi , where M Bi is the Banach space of elements iI

 f x  dx  B  S ,  f x  dx   , i

B

i

E



M Bi

  f x i dx  .

E

t  t1 , t2 ,...,tn   T , where T be closed bounded subset of R , t  T , i  I Gaussian real homogenous Ei t   0, i  I field on T with zero means and correlation function Let

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The Consistent Estimators For Homogeneous Gaussian Fields Statistical Structures

Ei t i S   Ri t  S , t , S  T , i  I . Let i , i  I  be the corresponding probability measures given

on S and f i  ,

  R n , i  I be spectral densities.

We be called the Fourier transformation of generation Fourier transformation.

Let

2 ~ bi , j  ,  

  f   f   dd  

Rn Rn

i

~ i, j  I , where bi , j ,  , ,   Rn , i, j  I the Generalization

i

S , t  T , i, j  I .

Fourier transformation of the following functions bi , j S , t   Ri S , t   R j S , t , S Then the corresponding probability measures

i and  j are pairwise orthogonal i, j  I (see [5]) and

E, S , i , i  I  are Gaussian orthogonal homogeneous fields statistical structures.Next, we consider Smeasurable g i  x , i  I function such that   g i  x  i dx    . Let M B the set measures defined by

 B     g i x i dx  ,

formula

iI E

I1  I

where

iI1 B

  g x   dx    , define a norm on iI1 E

i

i

a

countable

M B by formula    iI

subsets

 g x  dx , i

i

in

then

I

and

M B is a

E

M B   M Bi , where M Bi in Banach space of elements the form

Banach space of measures and

iI

 B    gi x i dx , B  S ,  gi x i dx   , with the norm on M B  i

iI1

E

M Bi

  gi x i dx .

E

It is also well known that in the (ZFC), (CH), (MA) theory there exists a weakly separable statistical structure which is not strongly separable.Here and in the equal we denote by (MA) the Martin’s axiom (see [2]-[8]).

Theorem 2. Let M B   M Bi be a Banach space of measures, E be the complete separable metric space. S be iI

the Borel  -algebra in E and cardI  c , where c denotes a power of continuum. Then in the theory (ZFC)&(MA) the Gaussian Homogeneous fields orthogonal statistical structures E, S , i , i  I  admits a consistent estimators of parameters i  I if and only if the correspondence

 f x  dx   l  , i

f

i

f  l f defined by the equality

i  M B is one-to-one. Here l f is a linear continuous functional on M B ,

E

f  F M B . Denote by F  F M B  the set of real functions f for which

Proof.Necessity. The existence of a consistent estimator

i  I implies that i i  I  i x :  x   i  1 .

Setting

2)

 : E, S   I , BI 

of the parameter

X i  x :  x   i for i  I , we get:

i  X i   i x :  x   1 for i  I ; X i1  X i2  

for

all

different

parameters i1 and

X  x :  x   i   x :  x   i   X   ;  X  x :  x  I   E. i1

3)

is defined

i

E

i  M B .

1)

 f x  dx 

1

2

i2

from

I

because

i2

i

iI

Therefore a statistical structure

X i , i  I

such that

E, S , i , i  I 

is strongly separable, so there exist S-measurable sets

1, if  j 0, if  j

i X j   

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The Consistent Estimators For Homogeneous Gaussian Fields Statistical Structures We put the linear continuous functional l X i into the correspondence to a function I X i  F M B  by the formula:

 I x  dx   l    Xi

i

I Xi

i

i

E

M Bi

.

l f1 into the correspondence to the function f1 x  f1 xI Xi x Then

We put the linear continuous functional for

any

i  M B i   f1 x i1 dx    f1 x I X i x i1 dx    f x  f1 x I X i x i dx  l f1 i1   i1 1

E

E

M Bi

E

.

 be the collection of extensions of functional satisfying the condition l f  px  on those subspaces

Let

where they are defined.

Let us introduce on  a partial ordering having assumed l f1  l f 2 if l f 2 is defined on large set then

l f1 and l f1  l f 2 there where both of them are defined.

l 

Let

fi iI

M iI

Bi

 . Let M Bi be the subspace on which l fi is defined. Define l f on

be a linear ordered subset in

having assumed l f    l fi   if

 MB . i

It is obvious, that l fi  l f . Since any linearly ordered subset in

 has an upper bound by virtue of Chorn’s

 contains a maximal element  defined on some set X  satisfying the condition  x   px  for x  X . But X  must coincide with the entire space M B because otherwise. We could extended  to a wider space by adding as above one more dimension. This contradicts the maximality of  hence X   M B . lemma

Therefore the extension of the functional is defined everywhere.The extension of the functional is defined everywhere.

It we put the linear continuous functional l f into correspondence to the function

f x    gi x I X i x   F M B 

then

 f x dx 

obtain

iI

   i iI

E

,

where

M Bi

 B     gi x i dx , B  S iI B

The necessity is proved. f F Sufficiency. For

M B  we

 f x dx l  . Denote

define

continuous

i1

i2

fi1

i2

1

2

i1

i1

E

i .

measure

1

Let

i1

fi  0

a.

e.

with

respect

i

i

i

i

i

i

c

j

1

2

j

i1

have

the

measure

i

and

j  i.

E

i

i

E

Hence it follows that

fi

i  I f . Let us

f i1  f1 a.e. with respect

therefore to

equality

i , i  M B

E

 f x  dx   ,  c    f x  dx , then  f x  dx   l    0 Denote Ci  x fi x  0, then  f  x  dx   0 j  0.

E

the

i

to which there corresponds. Then for

 f x  dx   l     f x  f x  dx    f x  dx 

by

 f x  dx   0 for E

l f i on M Bi

E

lf

functional

I f a countable subset in I, for which

f

consider functional

linear

 j Ci   0 j  i. On

the other hand

i E  Ci   0 therefore

the statistical

E, S , i , i  I  is weekly separable. A Boral orthogonal family of probability measures i , i  I , cardI  c is weakly separable. Represent i , i  I  as an inductive sequence i  w1 where

structure

w1 denotes the first ordinal number of the power of the set I.

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The Consistent Estimators For Homogeneous Gaussian Fields Statistical Structures Sense the statistical structure

E, S , i , i  I is

weakly

separable, there exists a family S-measurable

1, if i  j 0, if i  j

 

sets X i , i  I such that the following relations are fulfilled: i X j  





For all i  0, w1  and j  0, w1 

We define w1 sequence of parts Z i of the space E so that the following relations are fulfilled: 1) Z i is Borel subset in E for all i  w1 ; 2) Z i  X i for all i  w1 ; 3) Z i  Z j   for all i  w1 , j  w1 ,

i Z i   1 for all i  w1 .

4)

Assume that Z 0  X 0 . Let further the partial sequence



i  j;

Z 

j j i

be already defined for i  w1 .



    Z j   0 . Thus there exists a Borel subset Yi

It is clear, that

of the space E such that the following

 ji  relations are valid:  Z j  Yi and  Yi   0 . Assume Z i  X i  Yi , thereby the w1 sequence of Zi i  w 1 j i

disjunctive measurable subsets of space E is constructed. Therefore structure E, S , i , i  I  is strongly separable.

i Z i   1 . For all i  w1 . A statistical

The statistical structure E, S , i , i  I  is strongly separated there exists a family Z i iI of elements of

 -algebra S1   domi  such that: iI

1.

i Z i   1, i  I ;

2. Z i1  Z i2   for all different parameters i1 and i2 from I; 3.

Z

i

 E.

iI

For x  E, we put  x   i, where i is unique parameter from the set I for which x  Z i . The existence of such a unique parameter I can be proved by using conditions (2), (3).

Now let Y  BI  . Then

x :  x  Y    Zi .

x :  x   Y   domi  for

We have to show that

0

iY

each i0  I .

If i0  Y then

x :  x   Y    Zi  Zi

0

iY



Z . i

iY i0

On the other hand, from the validity of the condition that the statistical structure E, S , i , i  I  is strongly separated it follows that

 

Zi0  S1   domi   dom i0 . iI

On the other hand, the validity of the condition equality yields that

 Z dom . i

iY  i0

Since dom

Z

iY i0

i

   E  Z i0 implies that i0   Z i   0. The latter  iY i   0 



i0

  is  -algebra, we deduce that x :  x  Y   Z i0

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

i0



Z

iY i0

i

 

 dom i0 .

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The Consistent Estimators For Homogeneous Gaussian Fields Statistical Structures If i0  Y , then

x :  x  Y    Zi  E  Zi  iY

relation implies that x :   x   Y  dom

0

and we claim that

 . i0

iI

0

 

Thus we have shown the validity of the condition x :   x   Y  dom

x :  x Y  

i x :  x  Y   0. The letter

 

i0

for an arbitrary i0  I . Hence

dom i0  S1. We have shown that the map  : E, S   I , BI  is measurable

map.

Since BI  contain all singletons of I, we claim that i i  I  i x :  x  i  i Z i  1 .

Theorem 2 is proved.

Analogously is proved Theorem3. Let M B   M Bi Be a Banach space of measures.E be the complete metric space, whose

 



  

  

iI

topological weights are not measurable in a wider sense. S be the Borel  -algebra in E and cardI  c. Then in the theory (ZFC)&(MA) the Gaussian homogeneous fields orthogonal statistical structures E, S , i , i  I 

f  l f Defined by the

admits a consistent estimators of parameters i  I and only if the correspondence equality

 f x  dx   l   j

f

i

i  M B is one-to-one. Here l f is linear continuous functional on

E

M B , f  F M B  Denote by F  F M B  the set of real functions f for which

 f x  dx  is defined j

E

i  M B .

II. References [1].

[2].

[3].

[4].

[5].

[6].

[7].

[8].

[9].

[10].

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ISBN: 978-2-659-69689-3 Saarbrucken Deutschland 2015 G. Pantsulaia. On orthogonal families of probability measures (in Russian) Trans.GP1-8 (3501-pp 106-112) (1989) G. Pantsulaia. On separation properties for families of probability measures. Georg Math. J 10(2) pp. 335-342 (2003)

DOI: 10.9790/5728-1303010106

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