K-th Upper Record Values from Power Lomax Distribution

International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.org Volume 4 Issue 10 || December. 2016 || PP-41-43
www.ijmsi.org 41 | Page
K-th Upper Record Values from Power Lomax Distribution
Ibrahim B. Abdul-Moniem
Department of Statistics, Higher Institute of Management sciences in Sohag, Sohag, Egypt
ABSTRACT: In this paper, recurrence relations between the single and the product moments of the K-th
upper record values from Power Lomax distribution are obtained.
Keywords: K-th upper record values, Power Lomax distribution, recurrence relations.
I. INTRODUCTION
The probability density function (pdf) of the Power Lomax (PL) distribution is defined by (Rady et al. [1])
 
 
 
1
1 1
1 ; 0, , , 0 .
x
f x x x



     

 
 
 
    
 
 
(1.1)
The corresponding reliability (survival) function of PL distribution is given by,
   1 ; 0, , , 0 .
x
F x x


  


 
    
 
 
(1.2)
In view of (1.1) and (1.2), we get
     
1
.F x x x f x

  

  (1.3)
For more details on this distribution and its application one may refer to Rady et al. [1].
Let 1 2, , ..., nX X X be a sequence of independent and identically (iid) random variables with pdf  f x and
distribution function (df)  F x . For a fixed 1k  , we defined the sequence of K-th upper record times
 
 , 1
k
nU n  , of the sequence  1 2, , ..., nX X X as  
1
1
k
U  and
   
    : 11 : 1
m in : k k
n n
k k
n j j kn U U k
U J U X X   
  
For 1k  and 1, 2,...n  , we write  1
n nU U . Then  
 , 1
k
nU n  is the sequence of record times of
 1 2, , ..., nX X X . The sequence  
 , 1
k
nY n  , where  
 k
n
k
n
U
Y X is called the sequence of K-th upper
record values of  1 2, , ..., nX X X for convenience, we shall also take  
0
0
k
Y  . Note that for 1k  we have
 1
, 0
nn UY X n  which are record values of  1 2, , ..., nX X X (Ahsanullah, [2]).
The pdf of  
 1
k
nY n  is given by Dziubdziela and Kopocinski [3] as follows:
   
 
      
1 1
ln 1 1
1 !
k
n
n
n k
Y
k
f x F x F x f x
n
 
      
   

, (1.4)
and the joint pdf of    
, ; 1 , 2
k k
m nY Y m n n   is denoted by Grudzien [4] as:
     
   
            
1 1 1
,
, 1
1 ! 1 !
k k
m n
n
m n m k
Y Y
k
f x y H x H y H x h x F y f y
m n m
   
     
   
  
, (1.5)
where      
 
 
ln 1 an d
1
f z
H z F z h z
F z
    
 

,

www.ijmsi.org 42 | P a g e
Recurrence relations for single and product moments of k-th record values from some distributions are
established by many authors. From Weibull distribution is introduced by Pawlas and Szynal [5]. Saran and
Singh [6], discussed recurrence relations for single and product moments of k-th record values from linear
exponential distribution. Pawlas and Szynal [7], introduced recurrence relations for single and product moments
of k-th record values from Pareto, generalized Pareto and Burr distributions. Amin [8], introduced K-th upper
record values and their moments for uniform and Lomax distributions.
Relations for single and product moments of upper record values from Lomax distribution are introduced by
Balakrishnan and Ahsanullah [9].
II. SINGLE MOMENTS OF K-TH RECORD VALUES FROM PL DISTRIBUTION
In this section, recurrence relation for single moments of the K-th upper record values from PL distribution is
obtained. The single moments of K-th upper record values for PL distribution can be obtained from (1.4) (when
r is positive integer) as follows:
 
 
 
      
1 1
:
0
ln 1 1
1 !
n
n kr r
n k
k
x F x F x f x dx
n


 
      
   
  (2.1)
Lemma 2.1. For PL distribution, the following recurrence relation is satisfied.
 
 
 
 
 
 
 
 
 1 1 1 1
: : : 1 :
1
r r r r
n k n k n k n k
k
r
  
    
    

  

(2.2)
Proof. From (1.3) and (2.1), we get
 
 
 
 
 
        
 
    
1 11 1 1 1
: :
0
1
0
ln
1 !
ln
1 !
n n kr r r r
n k n k
n n k
r
k
x x F x F x f x d x
n
k
x F x F x d x
n
 
   
 

      


      
   
    
   


Integrating by parts, treating r
x for integration and the rest of the integrand for differentiation and after
simplification yields (2.2). □
Remarks
 When 1  , we get the recurrence relation for the single moments of K-th upper record values from
Lomax distribution, established by Amin [8].
 When 1  and 1k  , we get the recurrence relation for the single moments of upper record values from
Lomax distribution, established by Balakrishnan and Ahsanullah [9].
III. PRODUCT MOMENTS OF K-TH RECORD VALUES FROM PL DISTRIBUTION
In this section, recurrence relation for product moments of the K-th upper record values from PL distribution is
obtained. The product moments of K-th upper record values for PL distribution can be obtained from (1.5)
(when r and s are positive integers) as follows:
 
 
   
            
1 1 1,
, :
0
1
1 ! 1 !
n
m n m kr s r s
m n k
x
k
x y H x H y H x h x F y f y dy dx
m n m

 
   
     
   
    
(3.1)
Lemma 3.1. Let X be a non-negative random variable having a pdf (1.1), then
 
   
 
 
 
 , 1 , 1 , 1
, : , : , 1 :
1
, 1 2 an d , 0,1, 2, ...
1 1
r s r s r s
m n k m n k m n k
s
m n r s
k s k s
  
  
   
   


     
   
(3.2)
and
 
   
 
 
 
 , 1 , 1 , 1
, 1 : , 1 : :
1
, 1 an d , 0,1, 2, ...
1 1
r s r s r s
m m k m m k m k
s
m r s
k s k s
  
  
   
   
 

   
   
(3.3)
Proof.
For 1 2 and , 0,1, 2,...m n r s   
From (1.3) and (3.1), we have

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 
 
 
 
   
             
   
   
, 1 , 1
, : , :
1 1 11
0
1
0
1 ! 1 !
1
1 ! 1 !
n
r s r s
m n k m n k
m n m kr s
x
n
mr s
x
k
m n m
x y y y H x H y H x h x F y f y d y d x
k
x y H x H y
m n m


  

 
  
 
   
 

 
  
       
     
 
 
  
 
       
   
     
1
1
0
1
,
1 ! 1 !
n m k
n
mr
H x h x F y d y d x
k
x H x h x I x d x
m n m
 
 


    
   
 
 
   
where
        
1
1
n m ks
x
I x y H y H x F y dy

 
   
 
Integrating by parts, we get
                
2 111
1 1
1
n m ks
x
I x y H y H x F y k H y H x n m f y dy
s

  
           
     
 
This is implies that
 
 
 
 
 
 
 
 , 1 , 1 , 1 , 1
, : , : , : , 1 :
1
r s r s r s r s
m n k m n k m n k m n k
k
s
  
    
    

   
  
,
and after simplification yields (3.2). Proceeding in a similar manner for the case 1n m  , the recurrence
relation given in (3.3) can easily be established. □
Remarks
 When 1  , we get the recurrence relation for the product moments of K-th upper record values from
Lomax distribution, established by Amin [8].
 When 1  and 1k  , we get the recurrence relation for the product moments of upper record values from
Lomax distribution, established by Balakrishnan and Ahsanullah [9].
IV. CONCLUSION
In this paper, we introduce recurrence relations between the single and the product moments of the K-
th upper record values from Power Lomax distribution. Two special cases are obtained. These special cases are
recurrence relations between the single and the product moments of the K-th upper record values from Lomax
distribution introduced by Amin [8] and recurrence relations between the single and the product moments of an
upper record values from Lomax distribution established by Balakrishnan and Ahsanullah [9].
REFERENCES
[1] E. A. Rady, W. A. Hassanein, and T. A. Elhaddad, The power Lomax distribution with an application to bladder cancer data.
SpringerPlus 5:1838, 2016.
[2] M. Ahsanullah, Record Statistics, Nova Science Publishers, New York, 1995.
[3] W. Dziubdziela, B. Kopocinski, Limiting properties of the k-th record value, Appl. Math. (Warsaw) 15, 1976, 187–190.
[4] Z. Grudzien´ , Characterization of distribution of time limits in record statistics as well as distributions and moments of linear record
statistics from the samples of random numbers, PracaDoktorska, UMCS, Lublin, 1982.
[5] P. Pawlas, D. Szynal, Recurrence relations for single and product moments of k-th record values from Weibull distributions, and a
characterization, J. Appl. Statist. Sci. 10, 2000, 17–26.
[6] J. Saran, and S.K. Singh, Recurrence relations for single and product moments of k-th record values from linear exponential
distribution and a characterization, Asian J. Math. Stat. 1, 2008, 159–164.
[7] P. Pawlas and D. Szynal, Recurrence relations for single and product moments of k-th record values from Pareto, generalized Pareto
and Burr distributions, Comm. Statist. Theory Methods 28, 1999, 1699–1709.
[8] E. A. Amin, K-th Upper Record Values and their Moments. International Mathematical Forum, 6, 2011, 3013 - 3021
[9] N. Balakrishnan and M. Ahsanullah, Relations for single and product moments of record values from Lomax distribution, Sankhya
Ser. B 56, 1994, 140–146.

K-th Upper Record Values from Power Lomax Distribution

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