# Conditional Expectation of Generalized Order Statistics and Characterization of Probability Distributions

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Conditional Expectation of Generalized Order Statistics and Characterization of Probability Distributions
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J. Stat. Appl. Pro. Lett.  1 , No. 1, 9-18 (2014) 9 Journal of Statistics Applications & Probability Letters  An International Journal http://dx.doi.org/10.12785/jsapl/010102 Conditional Expectation of Generalized Order Statisticsand Characterization of Probability Distributions  Zubdah e Noor, Haseeb Athar  ∗ and Zuber Akhter  Department of Statistics and Operations Research, Aligarh Muslim University, Aligarh - 202 002, IndiaReceived: 20 May. 2013, Revised: 26 Sep. 2013, Accepted: 29 Sep. 2013Published online: 1 Jan. 2014 Abstract:  In this paper, two general form of distributions 1 − F  (  x ) =  e − ah (  x ) and 1 − F  (  x ) = [ ah (  x )+ b ] c ,  x  ∈  ( α  , β  )  are characterizedthrough the conditional expectation of power of difference of two generalized order statistics. Further, some of its important deductionsand particular cases are also discussed. Keywords:  Generalized order statistics, order statistics, record values, conditional expectation and characterization. 1 Introduction Kamps  introduced the concept of generalized order statistics ( gos ) to unify several models of ordered randomvariables, e.g. order statistics, record values, progressively type II censored order statistics and sequential order statistics.This common approach make it possible to deduce several distributional and moment properties at once. These modelscan be effectively applied in reliability theory and survival analysis.Therandomvariables( rvs ),  X  ( 1 , n , m , k  ) ,  X  ( 2 , n , m , k  ) , ...  X  ( n , n , m , k  ) , k  > 0 , m ∈ R are ngos fromanabsolutelycontinuousdistribution function ( df  )  F  (  x )  and probability density function (  pdf  )  f  (  x ) , if their joint density function is of the form k   n − 1 ∏  j = 1 γ   j  n − 1 ∏ i = 1 [ 1 − F  (  x i )] m  f  (  x i )  [ 1 − F  (  x n )] k  − 1  f  (  x n )  (1)on the cone  F  − 1 ( 0 )  <  x 1  ≤  ...  ≤  x n  <  F  − 1 ( 1 ) ,where  γ   j  =  k  +( m + 1 )( n −  j )  for all  j , 1  ≤  j  ≤  n ,  k   is a positive integer and  m  ≥ − 1.If   m  =  0 and  k   =  1, then  X  ( r  , n , m , k  ) , the  r  − th gos  reduces to the  r  − th  order statistics and (1) will be the joint  pdf   of order statistics  X  1: n  ≤  X  2: n  ≤  . . .  ≤  X  n : n  from  df F  (  x ) . If   m  =  − 1 and  k   =  1, then (1) will be the joint  pdf   of the ﬁrst  n upper record values. In view of (1), the  pd f   of   X  ( r  , n , m , k  ) , the  r  − th gos  is given by  f   X  ( r  , n , m , k  ) (  x ) =  C  r  − 1 ( r  − 1 ) ! [  ¯ F  (  x )] γ  r  − 1  f  (  x ) g r  − 1 m  ( F  (  x )) ,  α   ≤  x  ≤  β   (2)where, ¯ F  (  x ) =  1 − F  (  x ) and the joint  pdf   of   X  ( r  , n , m , k  )  and  X  ( s , n , m , k  ) , 1  ≤  r   <  s  ≤  n , is  f   X  ( r  , n , m , k  ) ,  X  ( s , n , m , k  ) (  x ,  y ) =  C  s − 1 ( r  − 1 ) ! ( s − r  − 1 ) ! [  ¯ F  (  x )] m  f  (  x ) g r  − 1 m  ( F  (  x )) × [ h m ( F  (  y )) − h m ( F  (  x ))] s − r  − 1 [  ¯ F  (  y )] γ  s − 1  f  (  y ) ,  α   ≤  x  <  y  ≤  β   (3) ∗ Corresponding author e-mail: haseebathar@hotmail.com c  2014 NSPNatural Sciences Publishing Cor.  10 Z. e. Noor et al: Conditional Expectation of Generalized Order Statistics... where, C  r  − 1  = r  ∏ i = 1 γ  i  ,  γ  i  =  k  +( n − i )( m + 1 ) , h m (  x ) =  −  1 m + 1  ( 1 −  x ) m + 1 ,  m   =  − 1 − log ( 1 −  x )  ,  m  =  − 1and  g m (  x ) = h m (  x ) − h m ( 0 ) ,  x ∈ ( 0 , 1 ) . The problem of characterization of distributions has always been the topic of interest among researchers. Variousapproaches are available in literature. Conditional expectation of ordered random variables are extensively used incharacterizing the probability distributions. Khan and Abu-Salih  have characterized some general form of  distributions through conditional expectation of function of order statistics ﬁxing adjacent order statistics. Khan andAbouammoh  extended the result of Khan and Abu-Salih  and characterized the distributions when the conditioning is not adjacent. Further, Samuel  characterized the distributions considered by Khan and Abu-Salih  for generalized order statistics ( gos ).Keseling  has generalized the result of Franco and Ruiz  in terms of generalized order statistics and characterized some general form of distributions. Khan  et al.   established characterizing relationship for the distributions throughgeneralized order statistics and characterized several distributions through conditional expectation of function of generalized order statistics.For more detailed survey on characterization one may refer to Franco and Ruiz [4,5], L ´opez-Bl´azquez andMoreno-Rebollo , Dembi´nska and Wesolowski [2,3], Wu and Ouyang , Khan and Athar , Wesolowski and Ahsanullah , Athar  et al.   and references there in.In this paper, an attempt is made to characterize two general forms of distributions  F  (  x ) =  1 − e − ah (  x ) and 1 − F  (  x ) =[ ah (  x )+ b ] c ,  x ∈ ( α  , β  )  through conditional expectation of   p − th  power of difference of functions of two generalized orderstatistics. 2 Characterization Let  X  ( r  , n , m , k  ) ,  r   =  1 , 2 , . . ., n  be  gos , then the conditional  pdf   of   X  ( s , n , m , k  )  given  X  ( r  , n , m , k  ) =  x , 1  ≤  r   <  s  ≤  n  inview of (2) and (3) is C  s − 1 C  r  − 1 ( s − r  − 1 ) ! [  ¯ F  (  y )] γ  s − 1 ( m + 1 ) s − r  − 1 [  ¯ F  (  x )] γ  r  + 1  [  ¯ F  (  x )] m + 1 − [  ¯ F  (  y )] m + 1  s − r  − 1  f  (  y ) . Theorem 2.1.  Let  X   be a random variable with an absolutely continuous df   F  (  x )  and pdf   f  (  x )  in the interval  ( α  , β  ) , where α   and  β   may be ﬁnite or inﬁnite, then for 1  ≤  r   <  s  ≤  n ,  E  [ { h (  X  ( s , n , m , k  )) − h (  X  ( r  , n , m , k  )) } 2 |  X  ( r  , n , m , k  ) =  x ] =  g r  , s  =  2! 1 a 2 s − 1 ∑ i 1 = r s − 1 ∑ i 2 = i 1 1 γ  i 1 + 1 1 γ  i 2 + 1 ,  (4)if and only if ¯ F  (  x ) =  e − ah (  x ) ,  a  >  0 ,  (5)where  h (  x )  be a continuous, differentiable and monotonic function of   x . Proof.  To prove the necessary part, we have  E  [ { h (  X  ( s , n , m , k  )) − h (  X  ( r  , n , m , k  )) } 2 |  X  ( r  , n , m , k  ) =  x ]=  C  s − 1 C  r  − 1 ( s − r  − 1 ) ! ( m + 1 ) s − r  − 1    β   x ( h (  y ) − h (  x )) 2  1 −   ¯ F  (  y ) ¯ F  (  x )  m + 1  s − r  − 1   ¯ F  (  y ) ¯ F  (  x )  γ  s − 1  f  (  y ) ¯ F  (  x ) dy .  (6) c  2014 NSPNatural Sciences Publishing Cor.  J. Stat. Appl. Pro. Lett.  1 , No. 1, 9-18 (2014) /  www.naturalspublishing.com/Journals.asp 11 Let  ¯ F  (  y ) ¯ F  (  x )  =  u , then  ( h (  y ) − h (  x )) 2 =  1 a 2  ( ln u ) 2 . Therefore,  RHS   of  (6) becomes =  1 a 2 C  s − 1 C  r  − 1 ( s − r  − 1 ) ! ( m + 1 ) s − r  − 1    10 ( ln u ) 2 ( 1 − u m + 1 ) s − r  − 1 u γ  s − 1 du .  (7)Now set  u m + 1 = t  , then (7) reduces to =  1 a 2 C  s − 1 C  r  − 1 ( s − r  − 1 ) ! ( m + 1 ) s − r  + 2    10 ( ln t  ) 2 ( 1 − t  ) s − r  − 1 t  γ  sm + 1 − 1 dt  .  (8)Since    10 ( ln u ) 2 u µ  − 1 ( 1 − u ) ν  − 1 d u  =  2  β  ( µ  , ν  ) ν  − 1 ∑ k  1 = 0 1 µ   + k  1 ν  − 1 ∑ k  2 = k  1 1 µ   + k  2 ( c.f.  Gradshteyn and Ryzhik, ; pp 543), where  β  ( µ  , ν  )  is complete beta function.Therefore, (8) becomes =  1 a 2 C  s − 1 C  r  − 1 ( s − r  − 1 ) ! ( m + 1 ) s − r  + 2 2 β  (  γ  s m + 1 , s − r  ) s − r  − 1 ∑ k  1 = 0 s − r  − 1 ∑ k  2 = k  1 1 γ  s m + 1  + k  1 1 γ  s m + 1  + k  2 =  2! 1 a 2 s − 1 ∑ i 1 = r s − 1 ∑ i 2 = i 1 1 γ  i 1 + 1 1 γ  i 2 + 1 . To prove the sufﬁciency part, let  E  [ { h (  X  ( s , n , m , k  )) − h (  X  ( r  , n , m , k  )) } 2 |  X  ( r  , n , m , k  ) =  x ] =  g r  , s or  C  s − 1 C  r  − 1 ( s − r  − 1 ) ! ( m + 1 ) s − r  − 1    β   x ( h (  y ) − h (  x )) 2  [  ¯ F  (  x )] m + 1 − [  ¯ F  (  y )] m + 1  s − r  − 1 [  ¯ F  (  y )] γ  s − 1  f  (  y ) dy =  g r  , s  [  ¯ F  (  x )] γ  r  + 1 . Differentiating both sides  w.r.t x , we get C  s − 1 C  r  − 1 ( s − r  − 2 ) ! ( m + 1 ) s − r  − 2    β   x ( h (  y ) − h (  x )) 2  [  ¯ F  (  x )] m + 1 − [  ¯ F  (  y )] m + 1  s − r  − 2 [  ¯ F  (  y )] γ  s − 1 [  ¯ F  (  x )] m  f  (  x )  f  (  y ) dy +  2 C  s − 1 C  r  − 1 ( s − r  − 1 ) ! ( m + 1 ) s − r  − 1 h ′ (  x )    β   x ( h (  y ) − h (  x ))  [  ¯ F  (  x )] m + 1 − [  ¯ F  (  y )] m + 1  s − r  − 1 [  ¯ F  (  y )] γ  s − 1  f  (  y ) dy =  γ  r  + 1  [  ¯ F  (  x )] γ  r  + 1 − 1  f  (  x ) g r  , s or γ  r  + 1  f  (  x ) ¯ F  (  x ) C  s − 1 C  r  ( s − r  − 2 ) ! ( m + 1 ) s − r  − 2    β   x ( h (  y ) − h (  x )) 2  [  ¯ F  (  x )] m + 1 − [  ¯ F  (  y )] m + 1  s − r  − 2 [  ¯ F  (  y )] γ  s − 1 [  ¯ F  (  x )] γ  r  + 2  f  (  y ) dy + 2  h ′ (  x )  C  s − 1 C  r  − 1 ( s − r  − 1 ) ! ( m + 1 ) s − r  − 1    β   x ( h (  y ) − h (  x ))  [  ¯ F  (  x )] m + 1 − [  ¯ F  (  y )] m + 1  s − r  − 1 [  ¯ F  (  y )] γ  s − 1 [  ¯ F  (  x )] γ  r  + 1  f  (  y ) dy =  γ  r  + 1  f  (  x ) ¯ F  (  x ) g r  , s c  2014 NSPNatural Sciences Publishing Cor.  12 Z. e. Noor et al: Conditional Expectation of Generalized Order Statistics... or γ  r  + 1  f  (  x ) ¯ F  (  x ) g r  + 1 , s  + 2  h ′ (  x ) 1 a s − 1 ∑  j = r  1 γ   j + 1 =  γ  r  + 1  f  (  x ) ¯ F  (  x ) g r  , s . After rearranging the above expression, we get  f  (  x ) ¯ F  (  x ) =  1 γ  r  + 1 2  h ′ (  x ) 1 a ∑ s − 1 i = r  1 γ  i + 1 [ g r  , s − g r  + 1 , s ] or  f  (  x ) ¯ F  (  x ) =  1 γ  r  + 1 2  h ′ (  x ) 1 a ∑ s − 1 i = r  1 γ  i + 1   2 a 2 ∑ s − 1 i 1 = r  ∑ s − 1 i 2 = i 1 1 γ  i 1 + 1 1 γ  i 2 + 1 −  2 a 2 ∑ s − 1 i 1 = r  + 1 ∑ s − 1 i 2 = i 1 1 γ  i 1 + 1 1 γ  i 2 + 1  =  1 γ  r  + 1 a h ′ (  x ) ∑ s − 1 i = r  1 γ  i + 1 1 γ  r  + 1 ∑ s − 1 i 2 = r  1 γ  i 2 + 1 , which implies,  f  (  x ) ¯ F  (  x ) =  ah ′ (  x ) . Hence the theorem. Remark 2.1.  Setting  m  =  0 and  k   =  1 in (4), Theorem 2.1 reduces for order statistics  E  [ { h (  X  s : n ) − h (  X  r  : n ) } 2 |  X  r  : n  =  x ] =  2! 1 a 2 s − 1 ∑ i 1 = r s − 1 ∑ i 2 = i 1 1 ( n − i 1 ) 1 ( n − i 2 ) if and only if  ¯ F  (  x ) =  e − ah (  x ) ,  x  ∈  ( α  , β  ) . Remark 2.2.  At  m  =  − 1 and  k   =  1, (4) reduces for upper record values  E  [ { h (  X  U  ( s ) ) − h (  X  U  ( r  ) ) } 2 |  X  U  ( r  )  =  x ] =  1 a 2  ( s − r  )( s − r  − 1 ) if and only if  ¯ F  (  x ) =  e − ah (  x ) ,  x  ∈  ( α  , β  ) . Lemma 2.1.  For any positive integers  µ   and  ν   with  n  ∈ N    10 ( ln u ) n ( 1 − u ) ν  − 1 u µ  − 1 du  = ( − 1 ) n n !  β  ( µ  , ν  ) ν  − 1 ∑ i 1 = 0 ν  − 1 ∑ i 2 = i 1 ··· ν  − 1 ∑ i n = i n − 1 1 µ   + i 1 1 µ   + i 2 ···  1 µ   + i n ,  (9)where  β  ( µ  , ν  )  is complete beta function. Proof.  Lemma can be established using the results of Gradshteyn and Ryzhik (; pp 540, 543). c  2014 NSPNatural Sciences Publishing Cor.  J. Stat. Appl. Pro. Lett.  1 , No. 1, 9-18 (2014) /  www.naturalspublishing.com/Journals.asp 13 Theorem 2.2.  Under the condition as stated in Theorem 2.1  E  [ { h (  X  ( s , n , m , k  )) − h (  X  ( r  , n , m , k  )) }  p |  X  ( r  , n , m , k  ) =  x ] =  g r  , s ,  p =  p ! 1 a  ps − 1 ∑ i 1 = r s − 1 ∑ i 2 = i 1 ··· s − 1 ∑ i  p = i  p − 1 1 γ  i 1 + 1 1 γ  i 2 + 1 ···  1 γ  i  p + 1 (10)if and only if ¯ F  (  x ) =  e − ah (  x ) ,  a  >  0 ,  (11)where  h (  x )  be a continuous, differentiable and monotonic function of   x . Proof.  To prove the necessary part, we have  E  [ { h (  X  ( s , n , m , k  )) − h (  X  ( r  , n , m , k  )) }  p |  X  ( r  , n , m , k  ) =  x ]=  C  s − 1 C  r  − 1 ( s − r  − 1 ) ! ( m + 1 ) s − r  − 1    β   x ( h (  y ) − h (  x ))  p  1 −   ¯ F  (  y ) ¯ F  (  x )  m + 1  s − r  − 1   ¯ F  (  y ) ¯ F  (  x )  γ  s − 1  f  (  y ) ¯ F  (  x ) dy .  (12)Let  ¯ F  (  y ) ¯ F  (  x )  =  u , then  ( h (  y ) − h (  x ))  p =  ( − 1 )  p a  p  ( ln u )  p . Therefore,  RHS   of  (12) becomes = ( − 1 )  p a  p C  s − 1 C  r  − 1 ( s − r  − 1 ) ! ( m + 1 ) s − r  − 1    10 ( ln u )  p ( 1 − u m + 1 ) s − r  − 1 u γ  s − 1 du .  (13)Setting  u m + 1 = t  , then (13) reduces to = ( − 1 )  p a  p C  s − 1 C  r  − 1 ( s − r  − 1 ) ! ( m + 1 ) s − r  +  p    10 ( ln t  )  p ( 1 − t  ) s − r  − 1 t  γ  sm + 1 − 1 dt  .  (14)Now using Lemma 2.1, we get = ( − 1 )  p a  p C  s − 1 C  r  − 1 ( s − r  − 1 ) ! ( m + 1 ) s − r  +  p  ( − 1 )  p  p !  β  (  γ  s m + 1 , s − r  ) × s − r  − 1 ∑ i 1 = 0 s − r  − 1 ∑ i 2 = i 1 ··· s − r  − 1 ∑ i  p = i  p − 1 1 γ  s m + 1  + i 1 1 γ  s m + 1  + i 2 ···  1 γ  s m + 1  + i  p =  p ! a  ps − 1 ∑ i 1 = r s − 1 ∑ i 2 = i 1 ··· s − 1 ∑ i  p = i  p − 1 1 γ  i 1 + 1 1 γ  i 2 + 1 ···  1 γ  i  p + 1 . Hence the (10). To prove the sufﬁciency part, let  E  [ { h (  X  ( s , n , m , k  )) − h (  X  ( r  , n , m , k  )) }  p |  X  ( r  , n , m , k  ) =  x ] =  g r  , s ,  p or  C  s − 1 C  r  − 1 ( s − r  − 1 ) ! ( m + 1 ) s − r  − 1    β   x ( h (  y ) − h (  x ))  p  [  ¯ F  (  x )] m + 1 − [  ¯ F  (  y )] m + 1  s − r  − 1 [  ¯ F  (  y )] γ  s − 1  f  (  y ) dy =  g r  , s ,  p  [  ¯ F  (  x )] γ  r  + 1 . c  2014 NSPNatural Sciences Publishing Cor.
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