Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 13, No. 1, 2024 

 

207 

Empirical	Bayes	Likelihood	for	Exponential	
Distribution	Family	under	Ranked	Set	Sampling	and	
Dependent	Data	
Naiyi Li1, Yongming Li2, Juan Huang1, * 

1School of Mathematics and Computer, Guangdong Ocean University, Zhanjiang 524088, China 
2School of Mathematics and Computer Science, Shang Rao Normal University Shang Rao, 334001, China 
* Corresponding Author: Juan Huang 

 

Abstract: In this paper, we get the empirical Bayes likelihood test rules for Exponential distribution based on ranked set 
sampling. Its asymptotic optimality is obtained. Due to the advantages of combining empirical Bayesian methods with prior 
information of samples, the accuracy of statistical inference can be improved. Therefore, the fusion of empirical Bayesian 
likelihood methods can further enhance the estimation methods of statistical models. 

Keywords: Ranked set sampling, empirical Bayes likelihood, Exponential distribution. 
 

1. Introduction 
Empirical Bayesian method is a method that uses historical 

samples to estimate prior distributions and determine 
Bayesian decision functions. This method is particularly 
useful when the prior distribution is unknown. Empirical 
Bayesian methods can usually be divided into two types: 
parametric empirical Bayesian methods and non parametric 
empirical Bayesian methods.In the parameter empirical 
Bayesian method, it is assumed that the prior distribution 
belongs to a known parameter family. Then, based on 
independent and identically distributed historical samples, 
classical statistical methods are used to provide estimates of 
parameters, thereby obtaining estimates of prior distributions. 
Using its Bayesian solution as an empirical Bayesian decision 
function.In non parametric empirical Bayesian methods, the 
Bayesian solution of a decision problem can be expressed as 
the product of two known functions. At this point, 
independent samples that follow a certain distribution can be 
used to estimate these two functions, thereby estimating the 
Bayesian solution and using it as the empirical Bayesian 
decision function for the decision problem. 

Overall, the empirical Bayesian method is a method that 
utilizes historical data to optimize decision-making. It 
combines prior information and sample information, obtains 
posterior information based on Bayesian formulas, and then 
infers unknown parameters based on the posterior 
information Ranked set sampling (RSS) was introduced by 
McIntyre to estimated pasture yields[1]. RSS has been used 
in many fields such as environment and sociology. Since 
Robbins proposed empirical Bayes (EB) approach, it has been 
developed [2-9].      

In this paper, we obtain empirical Bayes likelihood test rule 
for Exponential distribution based on ranked set sampling. 

Let X have a conditional density function for given θ  
 

       | expf x m x h x          (1.1) 

 
where θ is unknown parameter, 

 1 2x a x a        is the sample space and 

    10 | 0 xh m x e dx  


        is parameter 

space.  
We discuss the following test problem:  
 

0 0 1 0: :H H                  (1.2) 

 
Where H0 is given constants. 

We choose loss function 
     

     

0
0 0 0

0
1 1 0

, ,

, .

L d a I

L d a I

 
  


 

  



 


 

 

where 0 10, { , }a d d d   is action space, 0 1d and d  

imply acceptance and rejection of 0h  respectively.  

Suppose that the prior distribution ( )G   of parameter the 
θ is unknown.   

We have random decision function  
 

     δ(x)=P(accept 0H |X=x).            (1.3) 

 
Then, the risk function of δ(x) is shown by  
 

          

        

   

0 0

1 1

, , |

                  , | 1

                             G

R x G L d f x x

L d f x x dxdG

a x x dx C

    

   

 

 



 

 

 

 

   
where  
 

   1 1, ,GC L d dG 


   

          0 |x f x dG
 

  



  .     (1.4) 

 
The marginal density function of X is shown by 



 

208 

   

     

         

|G

x

f x f x dG

m x h e dG m x p x

 

 








 


   

 

where      xp x h e dG 


  . 

Since    1 z y

z
h e h e dy  


   ,we have  

 

         

       

1
| y

x

Gx

f x dG m x h e dydG

m x p y dy m x x

   




 

 





 

  


   

 

       1
,x

G x
x p y dy h e dG  


 


      (1.5)  

 

By (1.5), we have        0n n nx f x m x x     

where  
 

     
 

 

 
X xG

G x x

If y
x p y dy dy E

m y m x


   
     

 
  . 

 
Using (1.5), Bayes test function is obtained as follows  
 

 
 
 

1, 0,

0, 0.
G

x
x

x






 


 

 
Further, we obtian the minimum Bayes risk as follows 
 

     

   

inf , ,

.

G

G G

R G R G R G

a x x dx C


 

 


 

 
      (1.6) 

 

From above that    Gx x   and R(G) can be 

obtained when the prior distribution of  G   is given. If 

not, we use the EB method. The rest of this paper is organized 
as follows. Section 2 presents an EB test under ranked set 
sampling. In section 3, we obtain the optimal rate of 
convergence of the empirical Bayes likelihood test under 
ranked set sampling.  

2. Construction of Empirical Bayes 
Likelihood Test under Ranked Set 
Sampling and dependent data 

Supposed that X(1)1, X(1)2, ⋯, X(1)m, X(2)1, X(2)2⋯, X(2)m, ⋯, 
X(k)1, X(k)2⋯,X(k)m be a balanced ranked set sample from 
population which has the common marginal density function 
fG(x). We assume perfect ranking. Denote that X(1)1, X(1)m, X(2)1, 
X(2)m, X(k)1 ,X(k)m are ranked set historical samples, and X is 
present sample. Assume f(x)∈C𝑠,𝛼, x∈R1, where 
C𝑠,𝛼={g(x)|g(x) is a probability density function. The s−he 
order derivative Supposed that K(x) be a Borel measurable 
bounded function vanishing off (0,1) such that (A1):  

 
1

0

1, 01

0, 1, , 1!
t t
v K v dv

t st


   

 
      (2.1) 

 
empirical likelihood is established by 
 

1 11

( ( )) sup , 0, 1, 0
n n n

i i i i i
i ii

R f x np p p p
 

 
     

 
 

 
 
empirical likelihood ratio is established by 
 

1

( ( )) 2 log ( ( )) 2 log(1 )
n

i
i

l f x R f x s


    
. 

 

where ,s R s  is determined by  

 

1

1
( ) 0

1

n
i

i i

s
n s


  

 
. 

 
Kernel estimator of f(x) is defined by 
 

i =K(x−X(i)1/ℎ𝑛) ( )f x  

 
where ℎ𝑛 is a positive and smoothing bandwidth, and 

lim 0n
n
h


 , ( ) arg max ( ( ))nf x R f x . 

Thus, the estimator of ( )x  is shown by 

 

       0n n nx f x m x x            (2.2) 

 

And, the EB test function is defined as follows  
 

 
 
 

1, 0

0, 0

n

n

n

x
x

x






 


           (2.3) 

 

Let E stand for mathematical expectation with respect to 
the joint distribution of X(1)1, X(1)2, ⋯, X(1)m, X(2)1, X(2)2, ⋯, 
X(2)m, ⋯, X(k)1, X(k)2⋯, X(k)m. Then, the overall Bayes risk of 
δn is shown by  

 

      ,n n n GR x G a x E x dx C  


     
 

If      , , q
n GR G R G O n    , where 𝑞>0,  qO n  

is asymptotic optimalit convergence rates of empirical Bayes 

likelihood test function {  n x  }. Before proving the 

theorems, we need the following lemmas. Supposed that c,c1 
be different constants in different cases even in the same 
expression.   

Lemma. R(δ𝐺,G) and R(δ𝑛,G) are defined by above, then   
0≤R(δ𝑛,G)−R(δ𝐺,G)≤c∫Ω|β(x)|𝑃(|β𝑛(x)−β(x)|≥|β(x)|)dx.  



 

209 

3. Conclusion 
Theorem Assume (C1) and the following regularity 

conditions hold. where 
 1 2 s

nh n  ,  

(1)    1
x m x dx

 



  , 

(2)   1
x dx







  .  

Then, we get         1 2 2, , s s
n GR G R G O n         

Proof. Using Markov's inequality, we get  
 

   
     

     

       

1

1

1

0 , ,

 

 

     

 

n G

n

n G

n G

n n

R G R G

a x E x x dx

c x E f x f x dx

c x m x E x x dx

 

 

 

 

  



  













 

 

  



  





  (3.1) 

 

Applying Lemma and the conditions (1)-(2) in the Theorem, 
we obtain  

 
         11 2 2 1 2 2s s s s

n c n x dx c n
 

     


     (3.2) 

 

   12 2
n c n x m x dx c n

  
 


        (3.3) 

 
Substituting (3.2)-( 3.3) into (3.1),  

Remark 3.1. When 1   , s  ,     1 2 2s sO n   

nears  1 2O n . 

Acknowledgements 
This paper was financially supported by Natural Science 

Foundation of China (12161075), Natural Science 
Foundation of Guangdong Province (2022A1515010978, 
2024A1515011258) Natural Science Foundation of Jiangxi 
Province (20122ACB201006) and Natural Science 
Foundation of Guangdong Ocean University (R17083, 
C17201, P16091). 

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