Georgian Scientists/ქართველი მეცნიერები ტ. 5  N 1, 2023 308 

 

Georgian Scientists 

ქართველი მეცნიერები 

Vol. 5  Issue 1, 2023 

https://doi.org/10.52340/gs.2023.05.01.27 

 

On the accuracy by the metric 1L   of the density estimation constructed by dependent 

observations  

Beqnu Parjiani1, Levan Labadze2 and Tsiala Kvatadze3 

1Associate Professor of Department of Mathematics  of  Georgian  Technical  University; 2Invited Lecturer 

of Economics, Ilia State University; 3Invited Lecturer in Statistics, International Black Sea University 

Abstract 

A narrowly stationary two-component sequence  
1

,i i i
X


  is considered on the probability space 

 , ,F P . The control sequence  
1i i

  ( : ,  1,2,...)i i   is discrete  1 2, ,..., rb b b  , 

( )m miP b p  , 1,m r , 1,2,...i  , 
1

1
r

m

m

p


 .  
1i i

X


 ( : ,  1,2,...)iX R i   is a conditionally 

independent sequence, those members represent observations of some random variable X . The 

conditional distributions 
i mX b

, 1,m r  have unknown densities ( )mf x , 1,m r ,  respectively. A core 

Rosenblatt-Parzen-type estimate of the density is 
1

( ) ( )
r

m m

m

f x p f x


  constructed from the dependent 

observations. The accuracy of this estimate is determined by the metric 1L .  A special case obtained by 

using the Bartlett core and taking the smoothing coefficient as a specific sequence is considered. 

2010 Mathematics Subject Classification.  62G05.    62G07 

Introduction 

During statistical studies of practical tasks, parametric and non-parametric estimates are 

obtained. One important issue is the construction of an estimate of the density of the distribution. 

Until recently, estimates were made by independent observations. Many problems require 

consideration of dependent observations. 

Long-term financial independence studies of investment and insurance companies are 

constantly conducted in the financial market. It is necessary to assess the risks of banking investments. 

For this purpose, an analysis of the flow of reinvestments is carried out, and the indicators of the 

financial stability of the companies are evaluated. 



Georgian Scientists/ქართველი მეცნიერები ტ. 5  N 1, 2023 309 

The listed and other tasks require statistical analysis not only with independent, but also with 

dependent data. Research in this direction has actually just started. And there is a rich historical 

experience of constructing non-parametric estimates of density through independent observations. 

Let’s the quantities iX , ( iX R ), i = 1,2... represent independent observations of a random 

variable X . Let's say a quantity has an unknown density  g x . In M. Rosenblatt and E. In Parzen's 

works (see [1], [2]), the class of core estimations is considered as a density  g x estimation. 

    
1

ˆ , ,
n

n
n n n i

i

a
g x a k a x X

n 

          (1) 

where  
1n n

a


 is a sequence of positive numbers such that   

 lim , ,n n
n

a a o n


                    (2)  

and the core  K x  according to Lebesgue is a integrated certain Borel function. 

The accuracy of this type of density 2L approximation constructed by independent 

observations ([1], [3], [4]) and metrics ([5]) has been determined under different conditions. 

It is known by L. Devroye (see [5]) the accuracy of the metric 1L  of the estimate constructed  by 

independent observations of the density. Let's state this result as a lemma. We will use it during the 

proof of our theorem. 

Definition 1 (see [5]). Let us denote by F , the set of such functions  f x  (see [5]) that satisfy 

the conditions:  f x  is absolutely continuous and has a derivative almost everywhere, f  , f   are 

absolutely continuous and has a derivative almost everywhere, f  , f  are bounded and continuous. 

Definition 2 (see [5]). Let’s denote by   the set of such functions  x  (see [5]) that satisfy the 

conditions:  x  is a density with a compact carrier having derivatives up to the fourth order 

(including) F  , F  and    (1 )a x a x a   

Definition 3 (see [5]). Let’s denote by K  the class of densities bounded on R  with a compact 

carrier on (see [5]), for that  ( ) ( )K x K x   

Lemma (see [5]) Let’s the quantities iX  , ix R  , 1,2,...i  represent independent observations 

of some random quantity X  having an unknown density  g x  with a compact carrier. Let's say 

 ˆ ,n ng x a  is determined by the equality (1),  K x K   and na  is a sequence (2), then for the quantity 

 ˆ( ) , ( )n n nJ a g x a g x dx







   is fair the estimattion 

 
2

0

2 1
( ) ( ) sup ( )

2

n n
n a

an

a a
EJ a g x dx g x dx o

n a n

 



 

      
 

 


 


,   (3) 

where  

2 ( )K x dx





  ,  2 ( )x K x dx





  , 

  , and the symbol  - is a composition of functions. 



Georgian Scientists/ქართველი მეცნიერები ტ. 5  N 1, 2023 310 

If at the same time ( )g x F , then 

2

2 1
( ) ( ) ( )

2

n n
n

n

a a
EJ a g x dx g x dx o

n a n

 



 

 
    

 
 





.                            (4) 

Recently, statistical analysis of samples consisting of different types of dependent observations 

has start. 

We will construct a core Rosenblatt-Parzen-type estimation of the density with dependent 

observations. We consider conditionally independent observations. We will determine the accuracy of 

the built estimate with a metric 1L . 

On the probabilistic space (Ω, F, P) let us consider the two-component stationary (in the narrow 

sense) sequence of random variables  

 
1

,i i i
X


                                                                   ( 5) 

Where, : , : m

iX R  and      is some space. 

Definition 4. The sequence  
1i i

X


 in (5) is called a conditionally independent sequence (see. 

[6],[7],[8]) controlled by the sequence  
1i i

  if for any natural n  and the fixed trajectory 

 1 1 2, ,...,n n     the values 1 2, ,..., nX X X  become independent and for all natural numbers i , l , n , 

1 2, , , lj j j , 2 ;l n  i n ; 1 21 ... lj j j n      the equalities   

  1 1 1 2 21 2

1

, , ,
... ,

,

j j j j j jj j j n l ll

i ii n

X X XX X X

XX

  



   


 

are fulfilled, where  
X Y

 is the conditional distribution of the value X  under the condition Y . The 

conditionally independent sequence  
1i i

X


 in (3) is called a sequence with chain dependence (see. 

[9],[10],[11]) if  
1i i




 is a finite Markov chain with discrete time.  

We are considering the case, when the members of sequence 1{ }i i  are independent and 

identically distributed discrete random variables 

 1 2, ,..., rb b b   ;   1 i iP b p    , 1,i r  , 1 2 ... 1rp p p    . 

Remark: If the members of a sequence  
1i i

X


 represent elements of a statistical sample or 

observation, they are called conditionally independent (or, accordingly, chain dependent) 

observations. 

To determine the accuracy of estimates constructed with dependent observations, it is necessary 

to study the asymptotics of the sums of dependent random variables. In this process, the research 

methods of distribution of sums of independent random variables are is extended to dependent random 

variables. Markov dependence is considered in many practical and theoretical problems. It is one of 

the forms of weak dependence. Other forms of weak dependence are also known. In the works [6] and 

[9] are considered limit distributions of sums of conditionally independent random variables and limit 

theorems for functions defined on the Markov chain.  Many authors consider sums of random variables 

whose joint distribution is determined by some "control" sequence of random elements. I. Bokuchava, 

Z. Kvatadze and T. Shervashidze established limit distributions of normed sums for conditionally 



Georgian Scientists/ქართველი მეცნიერები ტ. 5  N 1, 2023 311 

independent random variables (see. [7], [8]) and for random variables with chain dependence ([10], 

[11]). 

The study of the asymptotic behavior of the sums of dependent random variables made it 

possible to consider dependent observations in the theory of statistical estimates. From the second half 

of the twentieth century, the construction of statistical estimates with dependent observations began. 

In this regard, the question of constructing a non-parametric estimate of the density on the dependent 

observations is particularly relevant. In the work of Yakowitz Sidney [12], estimates of density and 

regression coefficients are constructed from observations bound in a Markov chain. The accuracy of 

the density estimation constructed by  dependent observations by the  metric 2L  is known (see [13]). 

In the series of works [14-16], the Rosenblatt-Parzen-type kernel estimates of the density are 

constructed with chain dependence observations and conditionally independent observations. Their 

approximation accuracy with metrics 2L  and 1L is considered. Z. Kvatadze and B. Pharjiani's case 2r 

, the accuracy of the estimations constructed with both types of dependent observations was 

determined by metric 2L  (see [14]). In general, the estimate accuracy   constructed for observations 

with chain dependent is established 1L   (see [15]) and 2L   metrics (see [16]). 

Methodology  

During the proof of the theorem is applied the method I. Bokuchava, T. Shervashidze and Z. 

Kvatadze presented by in [10, 11]. Using this method, they determined the limit distributions of the 

normed sums of conditionally independent sequences (see [8]) and sequenge with chain dependence.  

The asymptotics of the conditional and unconditional distributions of the geometric mean of 

conditionally independent random variables and random variables with chain dependence were 

investigated (see [17], [18]). This method became possible to use in the theory of statistical estimations 

(see [14]). The method uses the decomposition of the sum to be estimated into sums corresponding to 

the values of the control sequence. On the fixed trajectory  1 1 2, ,...,n n     of the control sequence 

(5), the second components of the sequence (observations  
1i i

X


) become independent. The sums 

obtained from them are uncorrelated and consist of independent and uniformly distributed random 

variables.  

Main Results  

Let’s consider the sequence (5). i , 1, 2,.i  .., are independent and identically distributed 

discrete random variables. Let's assume that 

 1 2, ,..., rb b b   ;   1 i iP b p    , 1,i r  , 1 2 ... 1rp p p    . 

 Let us fix the trajectory  1 1 2, ,...,n n     of the sequence  
1i i




. In this case, we denote by the 

values  1n ,  2n ,…,  n r , the frequencies of accepting the values 1b , 2b ,..., rb   (respectively) by the 

members 1 , 2 ,..., n  of the sequence  
1i i




. 

   
1

I
k i

n

n b
k

i




 
 ,        1,i r , 

where  
I  is the indicator function. Obviously the equality is fair    



Georgian Scientists/ქართველი მეცნიერები ტ. 5  N 1, 2023 312 

     1 2 ...n n n r n      . 

Theorem. Let’s consider the sequence (5). Let’s say that members of the control sequence  
1i i




 

  1 2: , ,...,i rb b b   are independent, identically distributed, discrete random variables 

  1 2, ,..., rb b b  ,  1 k kP b p  , 1,k r , 1 2 ... 1rp p p    . Let us say that the sequence of 

positive numbers na  satisfies the conditions (2). Let's say that for each function 1: ,R   for that 

 iE     when noccurs the convergence  

                                   1( )
1

1 n
Ei

jn
    


      a. e.                             ( 6) 

Let’s say that the members of the sequence  
1i i

X


 are conditionally independent observations on some 

random variable X  and the conditional distributions 
1 1 iX b , 1,i r  have unknown densities ( )if x , 

1,i r  with a compact supports.   If inequalities  

 n i
i c

D
n n

 
 

 
 ,      1,i r  ,         (7) 

are fulfilled for frequencies  n i , 1,i r , then for each natural number n , the density estimate   

   
1

r

i i

i

f x p f x


  is the sum     
1

ˆ ,
n

n
n n n i

i

a
f x a K a x X

n 

  , where  K x K  , and for the value 

 ˆ( ) ( , )n n nJ a f x a f x dx





   the estimate  

1

2
( ) ( )

r
n

n i i

i

a
EJ a p f x dx

n



 

  


 

 
2 4

01 1

1
sup ( )

2

r r
n

i i a i

ai in

a
p f x dx c o

a nn






 

      
 

  ,   ( 8) 

 

is satisfied, where  

                              2K x dx





  ,      2x K x dx





  ,  a  .   

If also  if x F , 1,i r , then  

2 4
1 1 1

2 1
( ) ( ) ( )

2

r r r
n n

n i i i i i

i i in

a a
EJ a p f x dx p f x dx c o

n a nn






 

   

 
     

 
     (9) 

Proof. Of  Theorem. Let’s apply the method used in [10] [11]. Let's decompose the sum  ˆ ,n nf x a into r  

sums. Let's fix the trajectory  1 1 2, ,...,n n    . For each number i  (1 i r  ), we separately group the 

terms of those observations 1 2, ,..., nX X X out of the sum of  ˆ ,n nf x a , the corresponding 1 2, ,..., n  

control random variables of which took the value 9. Let's renumber the members of each sum.  

0 ( ) 0i  ,  1( ) min{ | ;  }m m i ii j j n b      ; 1,i r , 1, ( )nm i . 



Georgian Scientists/ქართველი მეცნიერები ტ. 5  N 1, 2023 313 

It turns out r  sequence of indices 

  1 2 ( )( ),  ( ),  ... ,  ( )
n ii i i   ,  1,i r . 

The equalities are fulfilled for them  

( ) ,    1, ,    1, ( )
m i i nb i r m i    . 

Now let's present the sum  ˆ ,n nf x a  as following  

1

( )ˆ ˆ( , ) ( , )
r

n
n n in n

i

i
f x a f x a

n




, 

where  
( )

( )

1

ˆ ( , ) ( ( )),   1, .
( )

n

m

i

n
in n n i

mn

a
f x a K a x X i r

i 

  





 

It is obvious that if ( ) 0n i  , then the corresponding sum  ,  does not exist. 

Let us show that are finite  and  values. 

Let's represent the value  ˆ ,n nEf x a  as a conditional mathematical expectation on the fixed 

trajectory 1n   

   
 

 1 1

1

ˆ ˆ ˆ, { ( , | )} { ( , | )}
r

n

n n n n n in n n

i

i
Ef x a E E f x a E E f x a

n


 



   . 

Let us take into account that, random variables ( )n i , ( 1,i r )  are commensurate with respect 

to the  -algebra induced by the partitioning of the space   generated when fixing the trajectory 1n  

(see [19]). Therefore, we can take them out of the determined by condition 1n  conditional 

mathematical expectation sign. Random variables ( ) ,   1, ( )
m i nX m i   are independent when the 

trajectory 1n  is fixed. They have the same conditional distribution 
1 1 iX b  with density ( )if x . 

 
  ( )

( ) 1

1 1

ˆ , { ( ( ( )) | )}
( )

n

m

ir
n n

n n n i n

i mn

i a
Ef x a E E K a x X

n i 

   








 

 
1 ( ) 1

1 1

( )
{ ( ( ) ( ( )) | )} ( ( )) ( ) ( )

( )

r r
n n n

n n i n n n i

i in

i a i
E E i K a x X a K a x u f u duE

n i n



  

     

 
 


 

Based on the condition (6), the equality 
 n

i

i
E p

n


  is fulfilled. Let's take into account that 

( )K x  is an even function and transform the variable ( )na u x t   under the integral sign. Will be 

obtained the equality 

 
1

ˆ , ( ) ( )
r

n n i i

i n

t
Ef x a p K t f x dt

a



 

   . 

( )K x  is a density and ( )if x  is a density bounded by a finite constant. Therefore,  ˆ ,n nEf x a  is a 

finite quantity. 

Let us show that the quantity  ˆ ,n nDf x a  is finite. 

      2

1
ˆ ˆ ˆ, { ([ , , ] | )}n n n n n n nDf x a E E f x a Ef x a    

 ˆ ,in nf x a 1,i r

 ˆ ,n nEf x a  ˆ ,n nDf x a



Georgian Scientists/ქართველი მეცნიერები ტ. 5  N 1, 2023 314 

 
 

 
  2

1

1 1

ˆ ˆ{ ([ , , ] | )}
r r

n n

in n in n n

i i

i i
E E f x a E f x a

n n

 


 

     

 
    2

1

1

ˆ ˆ{ ([ ( , , ] | )}
r

n

in n in n n

i

i
E E f x a Ef x a

n






   

On the fixed trajectory 1n , the quantities   ( )mn iK a x X 
, 1, ( )nm i , 1,i r  and 

accordingly, the sums  ˆ ,in nf x a , 1,i r  are independent.  

 
 

   2 2

1

1

ˆ ˆ ˆ, { ( ( ) [ , , ] | )
r

n

n n in n in n n

i

i
Df x a E E f x a Ef x a

n

  


  

 
   2 2

1

1

ˆ ˆ{( ) ([ , , ] | )}
r

n

in n in n n

i

i
E E f x a Ef x a

n






    

 

 
     

 
2 2

( ) ( ) 1

1 1

{( ) ([ ( )] | )}
n

m m

ir
n n

n i n i n

i mn

i a
E E K a x X EK a x X

n i 

     


 





 

 

 
  

 

  2 2 2

( ) ( ) 1

1 1

{( ) ( ) ( [ ] | )}
n

m m

ir
n n

n i n i n

m jn

i a
E E K a x X EK a x X

n i 

     


 





 

       
1 1

2
2

( ) ( ) 12
1

{ ( [ ] | )}
r

n
n n i n i n

i

a
E E i E K a x X EK a x X

n

         

 

     
 2

2

1

[ ( ) ] ( ) ( )
r

nn
n n i i

i

ia
K a x u K a x y f y dy f u duE

n n

 

  

     


 

Apply Equation 
 n

i

i
E p

n


  again. Let's apply the same variable transformation inside the 

integral sign as when considering the expression  ˆ ,n nEf x a . Will be obtained the equality 

     2

1

ˆ , [ ( ) ( ) ] ( )
r

n
n n i n i i

i n

a t
Df x a p K t K a x y f y dy f x dt

n a

 

  

      . 

Considering the properties of ( )K x  and equalities (2), it is clear that  ˆ ,n nDf x a  is finite. 

Let's estimate ( )nEJ a . 

1
ˆ( ) [ ( ( , ) ( ) | )]n n n nEJ a E E f x a f x dx





            

1

1 1

( ) ˆ[ ( ( , ) ( ) )]
r r

n
in n i i n i

i i

i
E E f x a p f x dx A

n



 

   


                    ( 10) 

 Each summand iA  is estimated in the same way.  

1

( ) ˆ[ ( ( , ) ( ) )]n
i in n i i n

i
A E E f x a p f x dx

n





  


  

1 1

( ) ( ) ( )ˆ[ ( ( , ) ( ) )] [ ( ( ) ( ) )]n n n
in n i n i i i n

i i i
E E f x a f x dx E E f x p f x dx

n n n

  
 

 

 

       



Georgian Scientists/ქართველი მეცნიერები ტ. 5  N 1, 2023 315 

1 1 1 2

( ) ( )ˆ{ [ ( , ) ( ) ]} { [ ( ) ]}n n
in n i n i i n i i

i i
E E f x a f x dx E E p f x dx A A

n n

 
 

 

 

        

  ˆ ,in nf x a  is a density estimate ( )if x  constructed from independent and identically distributed 

observations on a fixed trajectory 1n . To estimate the quantity
1

ˆ[ ( , ) ( ) ]}in n i nE f x a f x dx 




 , we apply 

inequality (3).   

1 1

( ) ˆ{ [ ( , ) ( ) ]}n
i in n i n

i
A E E f x a f x dx

n








  

 

 

 

( ) 1

1

{ [ ( ( )) ( ) | ]}
n

m

i

n n
n i i n

mn

i a
E E k a x X f x dx

n i














   

 

 
 

2
0

2 1
{ [ ( ) sup ( ) ( )]}

2 ( )

n n n
i a

an n n

i a a
E f x dx f x dx o

n i a i

 
 

  

 

 

      

According to the condition of theorem (6), the equations are fulfilled   

 n

i

i
E p

n


  , 

 
lim n

i
n

i
p

n




 . 

Therefore   ~n ii np  and accordingly we have equality. 

 
n n

n

a a
o o

i n

   
       

. 

The following assessment of the summand 1iA  is valid  

 
 1 2

0

2 ( ) 1 ( )
( ) { } sup ( ) ( )

2

n n n
i i a

an n

i a i
A f x dxE f x dxE

n i a n

  
 

 

 

 

    

( ) 2 ( )
( ) ( )n n n n

i

a i a i
o E f x dxE

n n n n

  







    
2

0

sup ( ) ( )
2

i n
a i

an

p a
f x dx p o

a n






 

   . 

Let's apply the inequality 
( ) ( )n n

i

i i
E E p

n n

 
  .   

 1 2
0

2
( ) sup ( ) ( )

2

n i i n
i i a i

an

a p p a
A f x dx f x dx p o

n a n


 



 

 

     . 

We take the value 
( )

( )n
i

i
p

n


  out of the sign of the conditional expectation (see [19]). Let's 

apply condition (6) and inequality (7). The following inequality will be obtained  

2

2 1

( ) ( ) ( )
[ ( ( ) | )] ( )n n n i

i i i n i i

i i i C
A E p E f x dx E p E p

n n n n

  






       . 

Finally, the following estimation of the summand iA  will be obtained  

 
2 4

0

2
( ) sup ( ) ( )

2

in i i n
i i a i

an

Ca p p a
A f x dxE f x dx p o

n a n n


 



 

 

      . 

We apply this estimation in inequality (10) and will be obtained the estimation of theorem (8). 



Georgian Scientists/ქართველი მეცნიერები ტ. 5  N 1, 2023 316 

The estimation of (9) is obtained directly from (8). Let's apply the following fact shown in [5] 

while proving the inequality of Lemma (4). For  g x F  class functions, the expression 

 
0

sup ( )a

a

g x dx



 

   does not depend on the selection of the  function  x  , and the equality is 

shown 

 
0

sup ( ) ( )a

a

g x dx g x dx

 

  

     

Corollary 1. If under the conditions  K x  of theorem is Bartlett's core 

 

Then for each natural number n  is the estimation of the density     
1

r

i i

i

f x p f x


  is presented by 

sum 

  2

1
[ ]

1

3
, (1 [ ( )] )

4 i
n

n
n

n n n i
x X

i a

a
f x a a x X

n  


     

and  for quantity  ( ) ( , )n n nJ a f x a f x dx





  , we have the estimate 

1

3 2
( ) ( )

5

r
n

n i i

i

a
EJ a p f x dx

n 



 

    

                     
2 4

01 1

0,1 1
sup ( ) ( )

r r
n

i i a i

ai in

a
p f x dx c o

a nn




 

      .             (11) 

If also  if x F , 1,i r , then the inequality is fulfilled 

2 4
1 1 1

3 2 0,1 1
( ) ( ) ( ) ( )

5

r r r
n n

n i i i i i

i i in

a a
EJ a p f x dx p f x dx c o

n a nn

 

   

        .         (12) 

Proof.  It is clear that    K x K x  ,  
3

4
K x   and  K x  have a compact support. Therefore, 

it satisfies the conditions of the theorem.  The inequalities (11) and (12) are obtained from (8) and (9) 

if we calculate   and   the quantities        

 
1

2 2 2

1

3 3
(1 )

4 5
K x dx x dx



 

      

 
1

2 2 4

1

3 1
( )

4 5
x K x dx x x dx



 

     . 

Corollary 2. If we apply the sequence na n  under the conditions of Corollary 1, we obtain 

the estimation of density    
1

r

i i

i

f x p f x


      

    2

[ 1]

3
(1 )

4
x

K x K x x


   



Georgian Scientists/ქართველი მეცნიერები ტ. 5  N 1, 2023 317 

  2

1
[ ]

1

3
, (1 [ ( )] )

4 i

n

n n i
x X

i n

f x a n x X
n  



     . 

The estimations are obtained from inequalities (11) and (12) 

4
1

1 6
( ) ( )

5

r

n i i

i

EJ a p f x dx
n 



 

    

 
4 4

01 1

0,1 1 1
sup ( ) ( )

r r

i i a i

ai i

p f x dx c o
n n n




 

      

4 4 4
1 1 1

1 6 0,1 1 1
( ) ( ) ( ) ( )

5

r r r

n i i i i i

i i i

EJ a p f x dx p f x dx c o
nn n n

 

   

        

Discussion 

Let's note that when proving the theorem, the trajectory of the control sequence

 1 1 2, ,..,n n     is fixed and the quantity ( )nEJ a  is presented as  1( ) ( ( ) )n n nEJ a E E J a  . This 

method makes it possible to use the independence of observations 1 2, ,..., nX X X  on a fixed trajectory. 

At this time, it becomes possible to expand the estimated sum by the method presented in [11] and 

[15]. Grouping of identically distributed values into one sum according to the values of the control 

sequence is used. Each sum is then represented as two sums of centered quantities. Estimations for one 

of them are written down using the methods used in [10] and [11]. The sums of the second type are 

evaluated by the classical results obtained in {5}. The measurability of the quantities  n i  and their 

compositions with continuous functions (see [19]) with respect to the sigma algebra generated by the 

division of the probability space is used when fixing the trajectory 1n . 

 

Conclusion. 

With the method used in the paper, it is possible to determine the exact upper boundaries of 

the obtained estimates. The method gives the possibility to be used in determining the accuracy of 

other (including non-parametric) estimates. 

We would like to be thankful to our colleague Prof. Tengiz Shervashidze and will honor his 

bright memory. 

 

References 

1. M. Rosenblatt (1956), Remarks on some nonparametric estimates of a density function. Ann. 

Math. Statist. 27, Chicago, 832-837. 

2. E. Parzen (1962), On estimation of a probability density function and mode. Ann. Math. 

Statist.33, Stanford, USA, 1065-1076. 

3. G. S. Watson, M.R. Leadbetter  (1963) On the estimation of the probability density. Ann. 

Math. Statist. 34, Toronto, : 480-491. 

4. E. A. Nadaraya (1983), Nonparametric estimation of the probability density and regression 

curve. (in Russian) Tbilisi State Univ. Press, 1983. 



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5. Devroye L., Györfi L. Nonparametric density estimation: the 1L  view. Wiley series in 

probability and mathematical statistics, Canada: John Wiley & Sons; 1985. p. 367. 

6. R. Z. Khasminski, On limiting distributions for sums of conditionally independent random 

variables. (Russian) Teor. Verojatnosti. i primenen. 6. (1961). №1,.119-125. 

7. G. L. O'Brien, Limit theorems for sums of chain-dependent processes. J. Appl. Probability 11 

(1974), 582-587. 

8. Bokuchava I. V. (1984) Limit theorems for conditionally independent sequences. . (in 

Russian) Teor. Verojatnost. i Primenen.-MathNet.Ru XXIX. (1984). №1,. p. 192-193.1984. 

Theory of probability and its applications, 1985, 29:1, 190-196 

9. Kvatadze Z., Shervashidze T. (2008) Some limit theorems for I.I.D. and conditionally 

independent random variables. The Second International Conference, “Problems of 

Cybernetics and Informatics“. September 10-12, 2008. Baku, Azerbaijan. Section #4. “Applied 

Stochastic Analysis” www.pci2008.science.az/4/12.pdf. Azerbaijan National Academy of 

Sciences. INSTITUTE OF INFORMATION TECHNOLOGY. Printing House of “Information 

Technology” Baku. 2008. Vol. II. pp.217-219. 

10. I. V. Bokuchava, Z. Kvatadze, T. Shervashidze, On limit theorems for random vectors 

controlled by a Markov chain. Probability theory and mathematical statistics, vol. I (Vilnius, 

1985), 231-250, VNU Sci. Press, Utrecht, 1987. 

11. Kvatadze Z., Shervashidze T. (1987) On limit theorems for conditionally independent 

random variable controlled by a finite Markov chain. Probability theory and mathematical 

statistics. (Proc. 5th Japan-USSR Symposium on Probability Theory. Kyoto. 1986). Lecture 

Notes in Mathematics, 1299: 250-259. Springer-Verlag. Berlin (Germany). 

12. Yakowitz Sidney (1989) Nonparametric density and regression estimation for Markov 

sequences without mixing assumptions. 85721–Journal of Multivariate Analysis, 30: 124-136. 

Arisona, USA. 

13. J. Meloche. Asymptotic behavior of the mean integrated squared error of kernel density 

estimators for dependent observations. Canadian Journal of Statistics. 2009., 18 (3): p. 205-

211. 

14. Z, Kvatadze, B. Phardjiani. On the Exsactness of Distribution Density Estimates Constructed 

by Some Class of Dependent Observations. Mathematics and Statistics. 2019 Vol. 7(4), pp. 

135-145.  SAN JOSE. 

15. Kvatadze Z., Pharjiani B., Construction of a kernel density estimator of Rosenblatt-Parzen   

type by conditionally independent observations. Proceedings of A. Razmadze Mathematical 

institute. Vol. 173. 2019. issue 3, 93-102. 

16. Pharjiani B., Kvatadze TS., Kvatadze Z., On an Application of Density Estimation 

Constructed by Means of Chain Dependent Samples.  Report of XXXIV enlarged session of 

the seminar of  I. Vekua Institute of Applied Mathematics (VIAM), of Ivane Javakhisvili 

Tbilisi State University (TSU). Sept. 16-19.  2020; Volume, 34. pp. 69-72. 

17. Kvatadze Z., Shervashidze T., Some limit theorems for IID and Conditionally independent 

random variables. The second international Conference, “Problems of Cybernetics and 



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Informatics”. September 10-12, 2008. Baku. Azerbaijan section N 4. “Applied Stochastic 

Analysis”. Science 14/12.pdf. Azerbaijan National Academy of Sciences. INSTITUTE OF 

INFORMATION TECHNOLOGY. Printing House of “Information Technology” Baku. 2008. 

Vol. II, 217-219. 

18. Kvatadze Z., Shervashidze T. On some Limit Theorems for sums and Products of Random 

Variables. Proceedings of A. Razmadze Mathematical institute. Vol. 150. 2009. 99-104. 

19. Kvatadze Z., Kvatadze TS. Limiting Distribution of a Sequence of Functions Defined on a 

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199-205. 

 

 

სიმკვრივის დამოკიდებული დაკვირვებებით აგებული   შეფასების  𝑳𝟏 მეტრიკით 

სიზუსტის შესახებ 

ბექნუ ფარჯიანი, ლევან ლაბაძე, ციალა ქვათაძე 

რეზიუმე 

თანამედროვე სტატისტიკურ კვლევებში მრავალი ამოცანის გადაჭრა, 

(როგორებიცაა მაგალითად საინვესტიციო და სადაზღვევო კომპანიების ფინანსური 

დამოუკიდებლობის კვლევები, საბანკო ინვესტიციების რისკების შეფასება, კომპანიების 

ფინანსური სტაბილურობის ინდიკატორების შეფასება და ა. შ.)  მოითხოვს 

დამოკიდებული დაკვირვებების განხილვას. დამოკიდებული მონაცემებით შეფასებების 

აგება ემყარება დამოუკიდებელი მონაცემებით შეფასებათა აგების მდიდარ ისტორიულ 

გამოცდილებას. აგებული შეფასებების სიზუსტის დადგენისთვის საჭიროა 

დამოკიდებული შემთხვევითი სიდიდეების ჯამების ასიმპტოტიკის შესწავლა. ამ 

პროცესში ხდება დამოუკიდებელ შემთხვევით სიდიდეთა ჯამების განაწილების 

კვლევის მეთოდების დამოკიდებულ შემთხვევით სიდიდეებზე გავრცობა. განიხილება 

მარკოვული დამოკიდებულება. ის სუსტად დამოკიდებულების ერთ-ერთი სახეა. 

მრავალი ავტორი იხილავს  ისეთ შემთხვევით სიდიდეთა ჯამებს, რომელთა ერთობლივი 

განაწილება განისაზღვრება შემთხვევით ელემენტთა რაიმე „მმართველი“ მიმდევრობით. 

ი. ბოკუჩავამ, თ. შერვაშიძემ და ზ. ქვათაძემ დაადგინეს პირობითად დამოუკიდებელი 

(Bokuchava I. V. (1984) Limit theorems for conditionally independent sequences. (in Russian) 

Teor. Verojatnost. i Primenen.-MathNet.Ru XXIX. (1984). №1, pp. 192-193.1984. Theory of 

probability and its applications, 1985, 29:1, 190-196; Kvatadze Z., Shervashidze T. (2008) Some 

limit theorems for I.I.D. and conditionally independent random variables. The Second 

International Conference, “Problems of Cybernetics and Informatics“. September 10-12, 2008. 

Baku, Azerbaijan. “Applied Stochastic Analysis” www.pci2008.science.az/4/12.pdf. Azerbaijan 

National Academy of Sciences. INSTITUTE OF INFORMATION TECHNOLOGY. Printing House 

of “Information Technology” Baku. 2008. Vol. II. pp. 217-219) და ჯაჭვურად დამოკიდებული 



Georgian Scientists/ქართველი მეცნიერები ტ. 5  N 1, 2023 320 

(I. V. Bokuchava, Z. Kvatadze, T. Shervashidze, On limit theorems for random vectors controlled 

by a Markov chain. Probability theory and mathematical statistics, vol. I (Vilnius, 1985), 231-250, 

VNU Sci. Press, Utrecht, 1987; Kvatadze Z., Shervashidze T. (1987) On limit theorems for 

conditionally independent random variable controlled by a finite Markov chain. Probability 

theory and mathematical statistics. (Proc. 5th Japan-USSR Symposium on Probability Theory. 

Kyoto. 1986). Lecture Notes in Mathematics, 1299: 250-259. Springer-Verlag. Berlin (Germany)) 

შემთხვევითი სიდიდეების ნორმირებული ჯამების ზღვარითი განაწილებები.   

არაპარამეტრულ შეფასებათა თეორიაში მნიშვნელოვანი ადგილი ეთმობა 

განაწილების უცნობი სიმკვრივის შეფასებას. მ. როზენბლატის და ე. პარზენის  

ნაშრომებში (M. Rosenblatt (1956), Remarks on some nonparametric estimates of a density 

function. Ann. Math. Statist. 27, Chicago, 832-837, E. Parzen (1962), On estimation of a 

probability density function and mode. Ann. Math. Statist.33, Stanford, USA, 1065-1076) 

განხილულია დამოუკიდებელი დაკვირვებებით აგებული სიმკვრივის გულოვანი 

შეფასებების კლასი. ცნობილია ამ შეფასებების სიზუსტე 2L   (E. A. Nadaraya (1983), 

Nonparametric estimation of the probability density and regression curve. (in Russian) Tbilisi 

State Univ. Press, 1983) და 1L  (Devroye L., Györfi L. Nonparametric density estimation: the 1L  

view. Wiley series in probability and mathematical statistics, Canada: John Wiley & Sons; 1985. 

p. 367) მეტრიკებით.  

დამოკიდებულ შემთხვევით სიდიდეთა ჯამების ასიმპტოტური ყოფაქცევის 

შესწავლამ შესაძლებელი გახადა სტატისტიკურ შეფასებათა თეორიაში დამოკიდებული 

დაკვირვებების განხილვა. ცნობილია სიმკვრივის არაპარამეტრული შეფასება და 

რეგრესიის კოეფიციენტების შეფასებები მარკოვის ჯაჭვად შეკრული დაკვირვებებით 

(Yakowitz Sidney (1989) Nonparametric density and regression estimation for Markov sequences 

without mixing assumptions. 85721–Journal of Multivariate Analysis, 30: 124-136. Arisona, 

USA). ასევე დადგენილია დამოკიდებული დაკვირვებებით აგებული სიმკვრივის 

შეფასების სიზუსტე 2L  მეტრიკით (J. Meloche. Asymptotic behavior of the mean integrated 

squared error of kernel density estimators for dependent observations. Canadian Journal of 

Statistics. 2009., 18 (3): p. 205-211). ზ. ქვათაძის და  ბ. ფარჯიანის მიერ აგებულია 

სიმკვრივის გულოვანი შეფასება პირობითად დამოუკიდებელი და ჯაჭვურად 

დამოკიდებული დაკვირვებებით და დადგენილია მათი სიზუსტე 2L  მეტრიკით იმ 

კერძო შემთხვევაში, როდესაც მმართველი მიმდევრობა იღებს ორ მნიშვნელობას ( 2r   ) 

(Z, Kvatadze, B. Phardjiani. On the Exsactness of Distribution Density Estimates Constructed by 

Some Class of Dependent Observations. Mathematics and Statistics. 2019 Vol. 7(4), pp. 135-145.  

SAN JOSE). ასევე მიღებულია ჯაჭვურად დამოკიდებული შეფასებებით აგებული 

გულოვანი შეფასების სიზუსტე  2L  მეტრიკით 2r   შემთხვევაშიც. 

წინამდებარე ნაშრომში პირობითად დამოუკიდებელი დაკვირვებებით 

აგებულია სიმკვრივის როზენბლატ−პარზენის ტიპის გულოვანი შეფასება. 

გამოყენებულია ი. ბოკუჩავას, თ. შერვაშიძის და ზ. ქვათაძის მიღებული შედეგები 

პირობითად დამოუკიდებელ შემთხვევით სიდიდეთა ჯამების ზღვარითი 



Georgian Scientists/ქართველი მეცნიერები ტ. 5  N 1, 2023 321 

განაწილების  შესახებ და დადგენილია აგებული შეფასების სიზუსტე 1L  მეტრიკით. 

თეორემის დამტკიცების დროს გამოიყენება თ. შერვაშიძის და ზ. ქვათაძის მიერ 

განხილული მეთოდი. დაფიქსირებულ  1 1 2, ,...,n n     ტრაექტორიაზე ხდება 

შესაფასებელი ჯამის დაშლა მმართველი მიმდევრობის მდგომარეობების შესაბამის 

ჯამებად. ეს ჯამები არაკორელირებულებია. დაფიქსირებულ  ტრაექტორიაზე 

თითოეული ჯამის შემადგენელი შესაკრებები დამოუკიდებლებია. გამოყენებულია 

ჯაჭვის მდგომარეობებზე განსაზღვრული ( )n i  ( 1,  i s ) ფუნქციების ზომადობა 

(Kvatadze Z., Kvatadze TS. Limiting Distribution of a Sequence of Functions Defined on a 

Markov Chain. Proceedings of A. Razmadze Mathematical institute. Vol. 174. 2020. issue 2, 

199-205) 1n  ტრაექტორიის დაფიქსირებით მიღებული ალბათური სივრცის დაყოფით 

ინდუცირებული   ალგებრის მიმართ. 

გამოყენებული მეთოდი  შესაძლებლობას იძლევა მიღებულ იქნას 

დამოკიდებული დაკვირვებებით აგებული სხვა ტიპის შეფასების სიზუსტე.