Journal of Modern Applied Statistical Methods
Volume 18 Issue 2 Article 23
9-17-2020
Concomitant of Order Statistics from New Bivariate Gompertz Distribution
Sumit Kumar Babasaheb Bhimrao Ambedkar University, [email protected]
M. J. S. Khan Aligarh Muslim University, [email protected]
Surinder Kumar Babashaheb Bhimrao Ambedkar University, [email protected]
Follow this and additional works at: https://digitalcommons.wayne.edu/jmasm
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Recommended Citation Kumar, S., Khan, M. J. S., & Kumar, S. (2019). Concomitant of order statistics from new bivariate Gompertz distribution. Journal of Modern Applied Statistical Methods, 18(2), eP2826. doi: 10.22237/jmasm/ 1604189820
This Regular Article is brought to you for free and open access by the Open Access Journals at DigitalCommons@WayneState. It has been accepted for inclusion in Journal of Modern Applied Statistical Methods by an authorized editor of DigitalCommons@WayneState. Journal of Modern Applied Statistical Methods November 2019, Vol. 18, No. 2, eP2826. Copyright © 2020 JMASM, Inc. doi: 10.22237/jmasm/1604189820 ISSN 1538 − 9472
Concomitant of Order Statistics from New Bivariate Gompertz Distribution
Sumit Kumar M. J. S. Khan Babasaheb Bhimrao Ambedkar University Aligarh Muslim University Lucknow, India Aligarh, India
Surinder Kumar Babasaheb Bhimrao Ambedkar University Lucknow, India
For the new bivariate Gompertz distribution, the expression for probability density function (pdf) of rth order statistics and pdf of concomitant arising from rth order statistics are derived. The properties of concomitant arising from the corresponding order statistics are used to derive these results. The exact expression for moment generating function (mgf) of concomitant of order rth statistics is derived. Also, the mean of concomitant arising from rth order statistics is computed using the mgf of concomitant of rth order statistics, and the exact expression for joint density of concomitant of two non-adjacent order statistics are derived.
Keywords: New bivariate Gompertz distribution, order statistics, concomitant of order statistics, moment generating function
Introduction
In the theory of concomitant of order statistics by David (1973), the values assumed by concomitant variable Y are corresponding to each value of variable X, arranged in some specified order, from a bivariate population (Xi, Yi); i = 1, 2,…, n. Concomitant of order statistics finds frequent application in selection problems, where decision of selection is based on the values of two random variables X and Y. For, e.g., X may be the score of students in preliminary examination and Y may be the corresponding score in final examination, X may be the height of an individual and Y may be the body weight of same individual in a certain physical examination test, etc.
doi: 10.22237/jmasm/1604189820 | Accepted: June 26, 2018; Published: September 17, 2020. Correspondence: M. J. S. Khan, [email protected]
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El-Sherpieny et al. (2013) discussed a new bivariate Gompertz distribution, denoted by NBG(α1, α2, α3, λ) which has Gompertz marginals. Univariate Gompertz distribution with parameters α, λ has the probability density function (pdf) is as follows:
−−(e1x ) f,,ee;,,0( xx) =x (1) and the cumulative distribution function (cdf) of univariate Gompertz distribution is given by
−−(e1x ) F,,1e( x ) =− . (2)
Suppose Ui ~ G(αi, λ), i = 1, 2, 3, are independently distributed. If we consider the random variable X = min(U1, U2) and Y = min(U2, U3), then the random vector (X, Y) shall follow NBG(α1, α2, α3, λ), α1, α2, α3, λ > 0, with corresponding joint cdf can be written as
F,;if1 ( x 0 yyx) F,F,;if( x yx )0= yxy 2 ( ) (3) F,;if3 ( x 0 yyx) = where
13+ x −−(e1) −−2 e1 y ( ) F,1ee1 ( xy) =−
23+ y −−1 e1x −−(e1) ( ) F,1ee2 ( xy) =−
1++ 2 3 x −−(e1) F,1e3 ( xy) =− and the corresponding joint pdf can be obtained as
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f,;if1 ( xyyx 0 ) f,f,;if( xyxyxy 0) = 2 ( ) (4) f,;if3 ( xyyx 0 ) = where
13+ x −−(e1) −−2 e1 y xy ( ) f,e1213( x eee ya) =+( )
23+ y −−1 e1x −−(e1) xy ( ) f,e2123( x eee ya) =+( )
123++ x −−(e1) x f,ee33( xy) =
Therefore, the marginal pdf and marginal cdf of X can be represented, respectively, as
13+ x−1 −(e ) x fee;0(xx) =+(13) , (5)
13+ x −−(e1) F1e(x) =− . (6)
Thus, the conditional pdf of Y given X is as follows:
f1 ( y | x) ; if 0 y x f( y | x) = f2 ( y | x) ; if 0 x y (7) f3 ( y | x) ; if y= x 0 where
22− e y y f12( yx |) = e e e
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23+ y −e 2 − 3 ex 23+ y f|eeee21( yx) = 13+
−−3 e1x 3 ( ) f|e3 ( yx) = 13+
Subsequently, Yang (1977) discussed the general distribution theory of the concomitants of order statistics. Bhattacharya (1984) also explained the theory and application of concomitant of order statistics in the name of induced order statistics. Balasubramanian and Beg (1996, 1997) obtained expressions of concomitant of order statistics in bivariate exponential distribution of Marshall and Olkin and concomitant of order statistics in Morgenstern type bivariate exponential distribution. Balasubramanian and Beg (1998) studied the concomitant of order statistics arising from Gumbel’s bivariate exponential distribution. Further, Begum and Khan (2000) and Begum (2003) obtained expressions of concomitant of order statistics from Marshall and Olkin's bivariate Weibull distribution and bivariate Pareto II distribution. In the same sequence of advancements to the theory of concomitant of order statistics, Thomas and Veena (2011) defined the application of concomitant of order statistics in characterizing a family of bivariate distributions. Chacko and Thomas (2011) and Philip and Thomas (2015) extended the application of concomitant of order statistics in the estimation of parameters of Morgenstern type bivariate exponential distribution and extended Farlie-Gumbel- Morgenstern bivariate logistic distribution, respectively. In this study, the th properties of concomitant of r order statistics, i.e. Y[r:n], are studied if the random variables (Xi, Yi), i = 1, 2,…, n, are iid and follows NBG(α1, α2, α3, λ).
Distribution of Order Statistics
Let X1, X2,…, Xn be the n iid random variables from the population having pdf f(x) th and cdf F(x). Then the pdf of r order statistics Xr:n is given by (David, 1981):
r−−1 n r fr:: n(x) = C r n F( x) 1 − F( x) f( x) ; − x , where Cr:n = n! / (r – 1)!(n – r)!. th Therefore, the pdf of r order statistics Xr:n, when X1, X2,…, Xn are iid random variables having pdf and cdf given in (5) and (6), respectively, is given by
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1313++x −−(n +− rn1e1 +) r ( ) feeexC=+ x r::13 nr( n ) ( ) r−1 (8) 1313++x e −1ee;0 x
Putting r = 1 in (8), we get the pdf of the first order statistics X1:n as
1++ 3 x 1 3 −nn e x f1:n (x) = n( 1 + 3 ) e e e ; 0 x . (9)
Let Xr:n and Xs:n be the two order statistics (1 ≤ r < s ≤ n); then the joint pdf of two order statistics is given by (David, 1981):
r−11 s − r − n − s fr,:12,: s n(x , x) = C r s n 1 − F( x 1) F( x 1) − F( x 2) F( x 2) f( x 12) f ( x ) ,
–∞ < x1 < x2 < ∞, where
n! Cr,: s n = (rsrns−−+−1) !1!!( ) ( ) and F̅ (x) = [1 – F(x)]. Therefore, if X1, X2,…, Xn are n iid random variables having pdf and cdf given in (5) and (6), respectively, then the joint pdf of Xr:n and Xs:n is given by
f,r, s : n (xx 1 2 ) =
r−11 s − r − 2 lk+ r−11 s − r − xx12 Cr, s : n ( 1+− 3 ) e e( 1) (10) lk==00 lk (+ ) −13(l +−− s r k)exx12 +−++( n s k 1) e −+−+( l n r 1) e
Distribution of Concomitant of Order Statistics
The pdf of concomitant of first order statistics Y[1:n] is given by (Begum & Khan, 2000)
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y gf|ff|ff|f( yyxxdxyxxdxyxx) =++ ( ) ( ) ( ) ( ) ( ) ( ) . 1:n y 11:21:31: nnn 0
Consider (7) and (9); the pdf of concomitant of first order statistics Y[1:n] is given by
nn(++) ( ) 221313 −−eeyx geeeeeeyndx=+ yx 1:n ( ) 213( ) y (+++) nyn( ) ( ) −−231313 eeeyxx 2 3 ++ndx eeeeeeeyx 123( ) 0 nn(++) ( ) 1313 −−−3 e1eyy ( ) y + n3eeee
After some algebraic simplification, we get
n ++ ( 1 3) 2 y (+ ) (1e− ) n + 231e− y yy 123( ) ( ) ge1:n ee( y) =+ e 2 n(133+−) (11) nn(1+ 3) −+ 21 + 3yy 3 ( ) (1−− e1) e ( ) n123( + ) yy −+e ee e n3 n(133+−)
The relation between the pdf of concomitant of first order statistics and concomitant of rth order statistics is given by (Balasubramanian & Beg, 1998)
n i− n + r −1 in−1 gr: n ( yy) =− ( 1) g 1: i ( ) . i= n − r +1 n− r i
Therefore, the pdf of concomitant of rth order statistics is given, for 0 < y < ∞, by
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i(++) n 132 1e− y i− n + r −1 in−1 ( ) y g1eern: ( y) =−( ) 2 nri− i= n − r +1 (+ ) i( + ) 23(1e− y ) + 123 ee y (12) i (133+−)
ii(132133+−++) yy( ) 1−− e1 e i( + ) ( ) ( ) −+123 eeeeyy i i (+−) 3 133
Moment Generating Function of Concomitant of Order Statistics
The moment generating function of concomitant of first order statistics Y[1:n] is defined as
ty MegY (tydy) = ( ) . 1:n 0 1:n
Thus, in view of (11), we have:
nn(1+ 3) ++ 21 + 3 2 ( ) y − e (ty+) MeeY (tdy) = e 2 1:n 0 + + ( 23) − 23e y n( + ) ty+ + 1 2 3 ee e ( ) dy n(1+− 3) 3 0 (13) nn(1+ 3) −+ 21 − 3 2 ( ) y − e n( + ) ty+ − 1 2 3 ee e ( ) dy n(1+− 3) 3 0
n(1++ 3) 2 n(1+− 3) 3 y − e + n e ee(ty+) dy 3 0
Consider the integral
n(1++ 3) 2 y −e I= ee(ty+) dy 1 0
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and
n(132++) 1 = ; after simplification, we get
1 tt I =+−+ Γ1γ1, , 11t +1 1 where Γ(.) is the gamma function and γ(., .) is the lower incomplete gamma function defined as
x γ,e(axudu) = au−−1 . 0
Thus, using the relationship between lower incomplete gamma function and Kummer’s confluent hypergeometric function, 1F1(1, s + 1, z), which is given by –1 s –z γ(s, z) = s z e 1F1(1, s + 1, z), (see Abramowitz and Stegun, 1972, p. 262),
−1 t +1 ttt −1 γ1,1eF+=++ 1,2,111 11 .
This implies
−1
ttt −1 Γ11+++ e F 1,2, 1 11 I =− . 1 t +1 1
Similarly, denoting
+ nn(+ ) −+ + ( ) ===23,, and 1 321 33 , 234
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respectively, and solving the rest of the integrals of (13), we get the mgf of concomitant of first order statistics given as
−1 ttt −1 Γ11eF+++ 1,2, 1 11 Me(t) =− 1 Y1:n 2 t +1 1 −1 ttt − Γ11eF+++ 1,2, 2 2 1 12 n( + )e +−123 t n +−+1 ( 133 ) 2 −1 ttt − Γ11eF+++ 1,2, 3 3 1 13 n( + )e − 123 − t n +− +1 ( 13) 3 3 −1 ttt −4 Γ11eF+++ 1,2, 1 14 +−n e 4 3 t +1 4 (14)
th Now, mgf of concomitant of r order statistics Y[r:n] can be obtained by utilizing the relation by Balasubramanian and Beg (1998):
n i− n + r −1 in−1 M1M tt=− . YYr:1: ni ( ) ( ) ( ) i= n − r +1 nri−
In view of (14), the mgf of concomitant of rth order statistics is given as
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M t = Yrn: ( ) −1 ttt −1 n Γ11eF1,2,+++ 1 11 i− n + r −1 in−1 −−1e 1 ( ) 2 t nri− +1 i= n − r +1 1 −1 tt− t Γ11eF1++2 ,2, + 2 112 i( + )e +−123 t i +− +1 ( 133 ) 2 −1 ttt − Γ11eF1,2,+++ 3 3 1 13 i( + )e −−123 t i +− +1 ( 133 ) 3 ttt −1 −4 Γ11eF1,2,+++ 1 14 +−i e 4 3 t +1 4
Mean of Concomitant of Order Statistics
th The p moment about origin of concomitant of first order statistics Y[1:n] can be obtained as
p p d YY= M (t) . 1:1:nn p dt t=0
Therefore, the first moment about origin i.e. mean for concomitant of first order statistics can be calculated as
d YY= M (t) . 1:nn 1: dt t=0
Using (14), we have
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−1 t t −1 t Γ+ 1 + 1 e1 F 1 1, + 2, 1 d =− e 1 Y1:n 2 t dt +1 1 −1 t t − t Γ+ 1 + 1 e2 F 1, + 2, 2 1 1 2 n( + )e +−1 2 3 t n +− +1 ( 1 3) 3 2 −1 t t − t Γ+ 1 + 1 e3 F 1, + 2, 3 1 1 3 n (+ )e − 1 2 3 − t n +− +1 ( 1 3) 3 3 −1 t t −4 t Γ+ 1 + 1 e1 F 1 1, + 2, 4 +−n e 4 3 t +1 (15) 4 t=0
Differentiating (15) with respect to t and using the relationships given below:
dxΓ ΓψΓxxx ==( ) , dx where ψ(x) is the digamma function (Prudnikov et al., 1990, p. 442), we have
dz− e−z 1F 12( 1;;F,;1,1; 2 +=+ zz) + + 2 + ( + + − ) , d (+ ) and setting t = 0,
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e1 ψ( 1log) − e−1 =+−+−211 F( 2,2;3,3;F 1;2;) ( ) Y1:n 2 211 11 1 4
2 n1232( +−)e ψ( 1log) 2 ++− 2F 22( 2,2;3,3; ) n( 133+−) 2 4 e−2 +−1F 12( 1;2; ) (16) 3 n1233( +−)e ψ( 1log) 3 −+− 2F 23( 2,2;3,3; ) n( 133+−) 3 4 e−3 +−1F 13( 1;2; )
4 −4 n3e ψ( 1log) − 4 4 e ++−+− 2F 241( 2,2;3,3;F 14 1;2;) ( ) 4 4
Now, the expression for pth moment about origin of concomitant of rth order statistics Y[r:n] can easily be obtained by using the relation (Siddiqui et al., 2011)
n i− n + r −1 in−1 pp=−1 . YYr:1: ni ( ) i= n − r +1 nri−
th In view of (16), the mean for concomitant of r order statistics Y[r:n] can be obtained as
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n 1 i− n + r −1 in−1 e ψ( 1log) − =−( 1) 2 1 Yrn: i= n − r +1 nri− 1
−1 1 e +−+−2F2,2;3,3;F1;2; 211( 11 ) ( ) 4
2 −2 i1232( +−)e ψ( 1log) 2e ++− 2F2,2;3,3; 22( ) i( 133+−) 2 4 e−2 +−1F1;2; 12( )
3 i123( + )e ψ( 1log) − 3 3 − +−2F2,2;3,3; 23( ) i( 133+−) 3 4 e−3 +−1F1;2; 13( )
4 −4 i3e ψ( 1log) − 4 4 e ++−+− 2F2,2;3,3;F1;2; 241( 14 ) ( ) 4 4
Similarly, differentiating mgf recursively with respect to t, the higher order moments may be obtained.
Joint Density of Concomitants of Two Order Statistics
th th If Y[r:n] and Y[s:n] are the concomitant of r and s order statistics, respectively, then the joint pdf of Y[r:n] and Y[s:n] is given by
x2 g,f|f|f, y yyxyxx= xdx dx , r,: s n ( 121122,) :1212 ( ) ( ) r s n ( ) − − where fr,s:n(x1, x2) is the joint pdf of two order statistics Xr:n and Xs:n and f(y1 | x1), f(y2 | x2) are the conditional pdf defined in (7). If the bivariate random variables (X, Y) follow NBG(α1, α2, α3, λ), then the joint density of concomitants of two order statistics Y[r:n] and Y[s:n] can be obtained by adding all joint densities existing at various mutually exclusive conditions, i.e.,
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g,,,,,r,: s n ( yyIyyIyyIyyIyy12112212312412) =+++( ) ( ) ( ) ( ) (17) +++IyyIyyIyy512612712( ,,, ) ( ) ( ) where I1(y1, y2),…, I7(y1, y2) are the joint densities under different mutually- exclusive conditions are and defined below.
Case I: when Y < X In this case, there are the following three sub-cases:
Sub-Case 1: If 0 ≤ y1 < x1 < y2 < x2 ≤ ∞.
y2 Iyxyxxxdx= f|f|f, dx . 1111222, :1212( ) ( ) r s n ( ) yy21
After some algebraic simplification, and in view of (7) and (10), we have
222yy r−11 s − r − −+ee12 2 lk+ rs−−11 r − 2 ( yy12+ ) ( ) IC1213,=+− : e ee1 ( ) r s n ( ) lk==00 lk
(++ ) −(l +++ n rl −11 s) −+− ry kn + + s k ( )( ) ( )( ) 1 31 31 3 2 −−eexx12 ee ee e xx12dx dx 12 yy21
Solving the above equation,
222yy r−11 s − r − −+ee12 lk+ r−11 s − r − 42 ( yy12+ ) ( ) IC12,=− : e e e1 r s n ( ) lk==00 lk +n − s + k +1 ( 13)( ) x1 (+ )(lnrlsrklsrk +−++ 1) − +−−+ +−−e ( )( ) ( )( ) 1 31 31 3 e eexx12− −eee (l+−− s r k )
Sub-Case 2: If 0 ≤ y1 < y2 < x1 < x2 ≤ ∞.
x2 I= f yx | f yx | f xxdxdx , . 2 111222,:12( ) ( ) r s n ( ) 12 yy22
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Again, after some algebraic simplification and in view of (7) and (10),
2 22−+eeyy12 32 ( yy12+ ) ( ) IC22,= : eee r s n (++) −(l + n r 1) 13 1e− y2 rs−−11 r − ( ) lk+ rsr−−−11e −( 1) lk==00 lk(lnrnsk+−+−++11)( )
Sub-Case 3: If 0 ≤ y2 < y1 < x1 < x2 ≤ ∞.
x2 Iyxyxxxdx= f|f|f, dx . 3111222, :1212( ) ( ) r s n ( ) yy11
Again, on using (7) and (10) and solving,
2 22−+eeyy12 42 ( yy12+ ) ( ) IC32,= : eee r s n (++) −(l + n r 1) 13 1e− y1 r−11 s − r − ( ) lk+ rs−−11 r − e −( 1) lk==00 lk(l+ n − rn +−11) s( + k + )
Case II: when X < Y Again, the three sub-cases are given as
Sub-Case 1: If 0 ≤ x1 < y1 < x2 < y2 ≤ ∞.
yy21 Iy= xyf|f|f, xx x dx dx . 41 112( 2) 2( ,:) 1212r s n ( ) y1 0
Using (7) and (10) and solving,
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++eeyy12 ( 23)( ) rs−−11 r − 2 − lk+ rsr−−−11 42 2 ( yy12+ ) I4123=+− ( ) eee1 ( ) lk==00 lk
(13++) −(l + n r 1) e1 (lsrknsk +−−+−−+++−)() ( 1)( ) 133133 (l+ s − r −+−+ kl)( s133133 − r −+− k) ( )( ) −−ex1 −ee
(n− s + k +1)(1 +−−33133) + ++− (n s k 1)( ) − eexx12− e − e
Sub-Case 2: If 0 ≤ x1 < x2 < y1 < y2 ≤ ∞.
yx12 Iyxyxx= f|f|f, xdx dx . 5111222, :1212( ) ( ) r s n ( ) 00
Again, on using (7) and (10) and solving,
yy12 (23++)(ee) 2 − 32 2 ( yy12+ ) I5=+ 1( 2 3 ) e e e
(13+)(l + n − r +1) r−11 s − r − lk+ r−11 s − r − e −( 1) lk==00 lk (l+ s − r − k )(1 + 3) − 3
(l+ s − r − k )(1 + 3) − 3 − (n− s + k +11)(1 + 3) − 3 ( n − s + k +)( 1 + 3) − 3 e −−e y1 −ee n− s + k +1 + − ( )( 1 3) 3 (n+ l − r +1)( + ) − 2 ( n + l − r + 1)( + ) − 2 1 3 3 1 3 3 y1 1 −−e −−ee n+ l − r +12 + − ( )( 1 3) 3
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Sub-Case 3: If 0 ≤ x1 < x2 < y2 < y1 ≤ ∞.
yx22 Iyxyxxxdx= f|f|f, dx . 6111222, :1212( ) ( ) r s n ( ) yy11
Again, using (7) and (10) and solving,
yy12 (23++)(ee) 2 − 32 2 ( yy12+ ) I6123=+ ( ) e ee
(13++) −(l + n r 1) r−11 s − r − lk+ rs−−11 r − e −( 1) lk==00 lk(l+ s − r −+− k )(133 )
(l+ s − r −+ k )(1 − 3) 3 − (n− s + kn ++11) s( k1 −− 3 +) ++ 31 3− 3( )( ) e −−e y2 −ee n− s + k ++−1 ( )( 133 ) (n+ l − rn ++1212) l( r −+ −) ++ − ( )( ) 1 331 33 y2 1 −−e −−ee n+ l − r ++−12 ( )( 133 )
Case III: When X = Y
Iyxyxxx7311322,= f|f|f,( :12 ) ( ) r s n ( ).
On using (7) and (10),
2 33−+eeyy12 2 ( yy12+ ) ( ) IC7= 3e e e r , s : n
r−11 s − r − (1+ 3)(l +−+ n r11) ( 1 + 3)( l +−− s r k) yy( 1 + 3 )( n −++ s k ) . lk+ r−11 s − r − −−ee12 −( 1) e e e lk==00 lk
Putting the values of I1,…, I7 in (17), obtain the required joint density of two concomitant of order statistics.
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References
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Prudnikov, A. P., Brychkov, I︠ U︡ . A., Marichev, O. I., & Gould, G. G. (1990). Integrals and series: More special functions (Vol. 3). New York: Gordon and Breach Science Publishers. Siddiqui, S. A., Jain, S., Siddiqui, I., Alam, M., & Chalkoo, P. A. (2011). Moments and joint distribution of concomitant of order statistics. Journal of Reliability and Statistical Studies, 4(2), 25-32. Retrieved from http://jrss.in.net/assets/4102.pdf Thomas, P. Y., & Veena, T. G. (2011). On an application of concomitants of order statistics in characterizing a family of bivariate distributions. Communications in Statistics – Theory and Methods, 40(8), 1445-1452. doi: 10.1080/03610921003606319 Yang, S. S. (1977). General distribution theory of the concomitants of order statistics. Annals of Statistics, 5(5), 996-1002. doi: 10.1214/aos/1176343954
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