Frontiers in Business, Economics and Management 
ISSN: 2766-824X | Vol. 14, No. 3, 2024 

 

231 

Empirical	Bayes	Likelihood	for	Distrubition	Family	
Based	on	Ranked	Set	Sampling	 	
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 empirical likelihood methods, the key to constructing a likelihood function is to find a suitable weight scheme 
that reflects the importance of data. These weights are usually limited by some constraints. By maximizing this likelihood 
function, we can obtain estimates of the parameters and perform corresponding hypothesis tests, In this paper, we get the 
empirical Bayes likelihood test rules for distribution based on ranked set sampling. Its asymptotic optimality is obtained. 

Keywords: Empirical Bayes likelihood, ranked set sampling. 
 

1. Introduction  
Empirical Likelihood Method is a statistical inference 

method aimed at constructing the 
likelihood function of data in a non parametric manner for 

parameter estimation and hypothesis testing. This method was 
initially proposed by Thomas and Grunkemeier and further 
developed by Owen. The main advantage of the empirical 
likelihood method is that it does not require any assumptions 
about the distribution of data, thus having robustness. [1-7]. 

Ranked Set Sampling (RSS) is a statistical sampling 
method that improves sampling efficiency to a certain extent, 
reduces experimental costs, and is particularly suitable for 
situations where samples are easy to sort but not easy to 
quantify[8]. 

The sorting set sampling method is divided into balanced 
sorting set sampling and unbalanced sorting set sampling. The 
process of balanced sorting set sampling is as follows: 
Randomly select samples with a capacity of the 2k  sample 
from the population and divide them into k groups, each 
containing k individuals.Sort each group of sample 
individuals in ascending order using visual or other intuitive 
information.Extract individuals from each group of samples 
arranged in order: select the individual with the smallest order 
from the first group of samples, select the individual with the 
second order from the second group of samples, and so on, 
until the individual with the highest order is selected from the 
k-th group of samples.After completing the above steps, 
samples with a capacity of k were selected from the 2k sample 
individuals, which is a complete cycle.  

The application examples of empirical Bayesian methods 
are indeed quite extensive. For example, in the field of signal 
processing, empirical Bayesian methods can be used for 
signal detection and classification. Through the prior 
probability and the conditional probability, we can calculate 
the posterior probability of different signals. These are just 
some examples of the application of empirical Bayesian 
methods. In fact, with the deepening of the research, the 
empirical Bayesian method also shows its potential and value 
in more fields. For example, in the field of machine learning, 
natural language processing, speech recognition and so on, 
empirical Bayesian methods have been widely used. It helps 
us understand and process data more accurately, leading to 

more informed decisions.empirical Bayes approach haved 
been stuied [9-14]     

In this paper, we obtain empirical Bayes likelihood test for 
distribution in ranked set sampling. 

Let X have a conditional density function for given θ  
 

1( | ) e (1 e )    x xf x               (1.1) 
 
where θ is unknown parameterand  0    is 

parameter space.  
We discuss the following test problem:  
 

0 0 1 0: :H H                (1.2) 
         

where 𝜃0 is given constants. 

We choose loss function      
2

0
0 0 0, ,L d I

 
  




 

     
2

0
1 1 0, .L d I

 
  




   

Where d d0,d1  is action space, d0 and d1 imply 
acceptance and rejection of 𝐻0 respectively.  

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

We have random decision function  
 

       δ(x)=P(accept 𝐻0|X=x).          (1.3) 
          

Then, the risk function of δ x  is shown by  

          
        

   

0

1

, , |

                  , | 1

                             G

R x G L d f x x

L d f x x dxdG

x x dx C

    

   

 

 



 

 

 

 



    

where  
 

   1,GC L d dG   ,        
2

0 |x f x dG      .(1.4)  
                   
The marginal density function of X is shown by 
 



 

232 

1( ) ( | )d ( ), ( ) e (1 e ) d ( ),    
      x xGf x f x G x G  

where  
 

(1) 2 2( ) e ( 1)(1 e ) d ( )x xG Gf f x G


   
       (1.5) 

 
By (1.5), we have (1)1 2( ) ( ) ( ) ( ) ( )G Gx u x f x u x f x     

where 21 ( ) e e 3
x xu x    , 2 ( ) u x  0(e 1)(e )

x 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

x x dx C


 

 


 

 
       (1.6) 

 
From above that δ x δG x  and R G  can be obtained 

when the prior distribution of G θ  is given. If not, we apply 
the empirical Bayes likelihood test method. The rest of this 
paper is organized as follows. Section 2 presents an empirical 
Bayes likelihood Test under ranked set sampling. In section 3, 
we obtain asymptotic optimality of convergence of the 
empirical Bayes likelihood test in ranked set sampling.  

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

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 thatX(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 𝑔(𝑠)(x) is continuous with 
|𝑔(𝑠)(x)|≤α,s≥3,α>0}, n km .  

Supposed that Kr(x) be a Borel measurable bounded 
function vanishing off (0,1) such that (A1):  

 

 
1

0

1, 01

0, 1, , 1!
t

r

t
v K v dv

t st


 

 
       (2.1) 

 
empirical likelihood is established by 

( )

1 11

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

r
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

r r
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 =Kr(x−X(i)1/ℎ𝑛)
( ) ( )rf x     

 where ℎ𝑛 is a positive and smoothing bandwidth, and 
lim𝑛→∞ℎ𝑛=0.  

( ) ( )( ) arg max ( ( ))
n

r rf x R f x  
Thus, the estimator of β(x) is shown by              
 

  (1)1 2( ) ( ) ( ) ( )n n nx u x f x u x f x         (2.2) 
 
And, the empirical Bayes likelihood 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)m,X(2)1, X(2)mX(k)1, X(k)m. 
Then, the overall Bayes risk of δn(𝑥) is shown by  

      ,n n GR x G a x E x dx C       
If lim𝑛→∞ R(δ𝑛,G)=R(δ𝐺,G),{δ𝑛(x)} is called asymptotic 

optimality of empirical Bayes likelihood test function.. 
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 [15]. R(δ𝐺,G) and R(δ𝑛,G) are defined by above, 
then   

0≤R(δ𝑛,G)−R(δ𝐺,G)≤c∫|β(x)|𝑃(|β𝑛(x)−β(x)|≥|β(x)|)dxΩ.  

3. Asymptotic Optimality of Empirical 
Bayes Likelihood Test in Ranked Set 
Sampling 

Theorem 3.1. Assume (A1) and the following regularity 
conditions hold. 

(1) ℎ𝑛>0, lim𝑛→∞ℎ𝑛=0,  

(2) ∫ 3 dG(θ)<∞,  
(3) 𝑓 (𝑥) is continuous function,   
Then, lim𝑛→∞ R(δ𝑛, G)=R(δ𝐺,G).  
Proof. Lemma 1 shows that 

0≤R(δ𝑛,G)−R(δ𝐺,G)≤𝑎∫|β(x)|𝑃(|β𝑛(x)−β(x)|≥|β(x)|)dxΩ.  
Applying Fubini theorem,  
we have 

3
0 0( ) )d ( 1) (3 )( 2) ( 1)( 2) ( )d ( ) dx x f x G x  

                   
2

0 0( 1) ( )d ( ) d (3 2 )( 2) ( )d ( ) df x G x f x G x   
             

( 1)( 2) ( )d ( ) df x G x
 

       
 

3
0 0( 1) (3 2 ) 1 d ( ) ( 1)( 2) d ( ) .G G 

                 

Denote  n x =|β(x)|𝑃(|β𝑛(x)−β(x)|≥|β(x)|).  
Obviously,  n x ≤|β(x)|. Then, by domain convergence 

theorem, we get  
 

0≤lim𝑛→∞ R(δ𝑛, G)−R(δ𝐺 ,G)≤∫[lim𝑛→∞  n x ]Ωⅆ𝑥. (3.1) 
 

Next, we need prove that lim𝑛→∞  n x =0 holds almost 
everywhere. By Markov's and Jensen's inequality,  

 



 

233 

 lim 0n
n

x


                   (3.2) 

 
Substituting (3.2) into (3.1), the proof of theorem 3.1 is 

finished.  

Acknowledgment 
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 (20122ABC201006) and Natural Science 
Foundation of Guangdong Ocean University (R17083, 
C17201,P16091). 

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