Concomitants of generalized order statistics from bivariate Lomax distribution

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1 ProbStat Forum, Volume 6, October 23, Pages ISSN ProbStat Forum s an e-ournal. For detals please vst Concomtants of generalzed order statstcs from bvarate Lomax dstrbuton Nayabuddn Abstract. In ths paper probablty densty functon (pdf for rth, r n and the ont (pdf of rth and sth, r < s n, concomtants of generalzed order statstcs from bvarate Lomax dstrbuton s obtaned. Also sngle and product moments are derved. Further the results are deduced for moments of kth upper record values and order statstcs. Also ther means and product moments are tabulated.. Introducton The Lomax dstrbuton ntroduced and studed by Lomax (954. He used ths dstrbuton to analyze busness falure data. Lomax dstrbuton has been studed by several authors n lterature. Balkema and de Haan (974 showed that the (df of Lomax dstrbuton arses as a lmt dstrbuton of resdual lfetme at great age. Accordng to Arnold (983, the Lomax dstrbuton s well adapted for modelng relablty problems. Nayak (987 used multvarate Lomax dstrbuton n relablty theory. Balakrshnan and Ahsanullah (994 derved the relatons for sngle and product moments of record values from Lomax dstrbuton. The Lomax dstrbuton s also known as the Pareto dstrbuton of second knd. In ths paper, we consder the bvarate Lomax dstrbuton (Sankaran and Nar, 993 wth probablty dstrbuton functon (pdf f(x, y α α 2 c(c + ( + α x + α 2 y (c+2, x, y, c, α, α 2 >, ( and correspondng df F (x, y ( + α x + α 2 y c, x, y, c, α, α 2 >. The condtonal pdf of Y gven X s f(y x α 2(c + ( + α x (c+ ( + α x + α 2 y (c+2, y >. (2 The margnal pdf of X s f(x cα, x >, (3 ( + α x (c+ Keywords. Generalzed order statstcs, concomtants of generalzed order statstcs, order statstcs, record values, bvarate Lomax dstrbuton, sngle and product moments Receved: 9 August 22; Revsed: 3 May 23; Re-revsed: 2 September 23; Accepted: 7 September 23 Emal address: nayabstats@gmal.com (Nayabuddn

2 and the margnal df of X s Nayabuddn / ProbStat Forum, Volume 6, October 23, Pages F (x ( + α x c, x >. (4 The concept of generalzed order statstcs (gos was gven by Kamps (995. Several authors utlzed the concept of (gos n ther work for detaled survey one may refer to Khan et al. (26, Ahsanullah and beg (26, Anwar et al. (27, Beg and Ahsanullah (28, Fazan and Athar (28, Tavangar and Asad (28, Khan et al. (29, Tahmaseb and Behboodan (22, Athar et al. (22, Athar et al. (23, Athar and Nayabuddn (23, Athar and Nayabuddn (23, among others. Let n N, n 2, k >, m (m, m 2,..., m n R n, M r n r m, such that γ r k + n r + M r > for all r {, 2,..., n }. Then X(r, n, m, k, r {, 2,..., n} are called gos f ther ont pdf s gven by k ( n ( n γ [ F (x ] m f(x [ F (xn ] k f(xn (5 on the cone F ( < x x 2... x n < F ( of R n. Choosng the parameters approprately, models such as ordnary order statstcs (γ n + ;, 2,..., n,.e. m m 2... m n, k, kth record values (γ k,.e. m m 2... m n, k N, sequental order statstcs (γ (n + α ; α, α 2,..., α n >, order statstcs wth non-ntegral sample sze (γ (α + ; α >, Pfefers record values (γ β ; β, β 2,..., β n > and progressve type II censored order statstcs (m N, k N are obtaned (Kamps, 995, 2. In vew of (5 wth m m;, 2,..., n, the pdf of rth gos, X(r, n, m, k s f X(r,n,m,k C r (r! [ F (x] γr f(xg r m (F (x (6 and ont pdf of X(s, n, m, k and X(r, n, m, k, r < s n, s where and f X(r,s,n,m,k (x, y C r C s (r!(s r! [ F (x] m f(xgm r ( ( ( F (x [hm F (y hm F (x ] s r [ F (y] γs f(y, α x < y β, (7 r γ, γ k + (n (m +, ( xm+, h m (x m + log( x g m (x h m (x h m (, x (,. m, m Let (X, Y,, 2,..., n, be n pars of ndependent random varables from some bvarate populaton wth dstrbuton functon F (x, y. If we arrange the X varates n ascendng order as X(, n, m, k X(2, n, m, k X(n, n, m, k, then Y varates pared (not necessarly n ascendng order wth these generalzed ordered statstcs are called the concomtants of generalzed order statstcs and are denoted by Y [,n,m,k], Y [2,n,m,k],..., Y [n,n,m,k]. The pdf of Y [r,n,m,k], the rth concomtant of generalzed order statstcs s gven as g [r,n,m,k] f Y X (y xf X(r,n,m,k (xdx (8

3 Nayabuddn / ProbStat Forum, Volume 6, October 23, Pages and the ont pdf of Y [r,n,m,k] and Y [s,n,m,k] s g [r,s,n,m,k] (y, y 2 x2 f Y X (y x f Y X (y 2 x 2 f X(r,s,n,m,k (x, x 2 dx dx 2. (9 It s well known that the dstrbuton functon of order statstcs are connected by the relatons (Davd, 98 n ( ( n F r:n (x ( n+r F : (x n r n r+ where F r:n (x s the df of rth order statstcs. Thus the pdf of rth concomtants of order statstcs Y [r:n] s g [r:n] (y n n r+ and the kth moments of Y [r:n] s µ k [r:n] (y n n r+ ( ( n ( n+r g [:] (y n r ( ( n ( n+r µ (k n r [:](y. ( Here some mportant transformaton and formulas are presented, whch wll be used n the subsequent sectons (Prudnkov et al., 986; Srvastava and Karlsson, 985 (λ n ( + z a ( n ( λ n, n, 2, 3,..., λ, ±, ±2,..., ( ( p (a p z p, (2 p! p (λ + m λ(λ + m (λ m, where (λ m Γ(λ+m Γ(λ, λ,, 2,... (3 (λ + m + n λ(λ + m+n (λ m+n, (4 (λ m+n (λ m (λ + m n. Important denttes/result n hypergeometrc functon are a, b 2F ; z (a p (b p ( z p (c c p p p! (5 (6 s condtonally convergent for z, z f < Re(ω, 2F a, b ; Γ(cΓ(c a b, Re(c a b >, c,,, 2,..., (7 Γ(c aγ(c b c a, b x p 2F ; mx dx (m p Γ(cΓ(pΓ(a pγ(b p, (8 Γ(aΓ(bΓ(c p c

4 Nayabuddn / ProbStat Forum, Volume 6, October 23, Pages f < Re p < a, Re b; arg m < π (a, (a 2, (a 3 x s 3F 2 ; x dx Γ(b Γ(b 2 Γ(sΓ(a sγ(a 2 sγ(a 3 s, (9 Γ(a (b, (b 2 Γ(a 2 Γ(a 3 Γ(b sγ(b 2 s f < Re < s < Re a ;, 2, 3 3F 2 N,, a m,, 2 b, (b (a m F 2, (b + N (a b, a m, 3 2 b N, 2 a, (2 N,, N, m, 3F 2, (l F 2,, (N + l l, m 3 m, 2 N l (2 for m, 2,..., l N, N, N,... For real postve k, c and a postve nteger b b ( b ( a B(a + k, c B(k, c + b. (22 a a Note that (Erdély et al., 954 y x λ (x y µ Γ(λ µγµ dx y (µ λ, < Reµ < Re < λ Γλ (23 µ, v x v (a + x µ (x + y ρ ΓvΓ(µ v + ρ dx Γ(µ + ρa µ y (v ϱ 2F ; y a µ + ρ (24 f arg a < Π, Rev >, arg y < Π, Reρ > Re(v µ. Note that (Srvastava and Karlsson, 985 F p:q;k (a p; (b q ; (c k p l:m;n ; x, y (a q r+s (b k r (c s x r l (α l ; (β m ; (γ n ; r s (α m r+s (β n r (γ s r! s known as Kampé de Féret s seres. Note that (Prudnkov et al., 986 pf q α, α 2,..., α p ; z β β 2,..., β q k (α k (α 2 k...(α p k (z k (β k (β 2 k...(β q k k! y s s! (25 (26 s known as generalzed hypergeometrc seres. 2. Probablty densty functon of Y [r,n,m,k] For the bvarate Lomax dstrbuton as gven n (, usng (2, (3, (4 and (6 n (8, the pdf of rth concomtants of gos Y [r,n,m,k] s gven as g [r,n,m,k] (y α 2C r c(c + r ( r (r!(m + r ( α ( + α x + α 2 y c+2 dx. (27 ( + α x c(γr

5 Nayabuddn / ProbStat Forum, Volume 6, October 23, Pages Let t α x, then the R.H.S. of (27 reduces to α 2 C r (r!(m + r r c(c + Usng relaton (24 n (28, we get g [r,n,m,k] (y ( r ( ( + t + α 2 y (c+2 ( + t c(γr dt. (28 C r (r!(m + r c(c + (α 2 r ( ( + α 2 y (c+ ( (cγ r r c, (cγ r + 2 F ; α 2 y. (29 (cγ r + 2 We now prove that g [r,n,m,k] (ydy. We have, g [r,n,m,k] (ydy Usng relaton (6 n (3, we get C r (r!(m + r r c(c + α 2 ( + α 2 y (c+ 2 F α 2C r c(c + r ( r ( (r!(m + r cγ r + Let t + α 2 y, then R.H.S. of (3 becomes C r c(c + r (r!(m + r ( r ( Now usng relaton (23 n (32, we get C r (r!(m + r r c(c + p cγ r + ( r ( ( r ( (cγ r c, (cγ r + 2 (cγ r c p ( p ( p (cγ r + 2 p p! p cγ r + cγ r + (cγ r c p ( p ( p (cγ r + 2 p p! p ; α 2 y dy. (3 ( + α 2 y (c+ (α 2y p dy. (3 t (c+ (t p dt. (32 (cγ r c p ( p ( p Γ(c p. (cγ r + 2 p Γ(c + In vew of (, we have C r (r!(m + r r (c + ( r ( cγ r + 3 F 2 (cγ r c,, (cγ r + 2, ( c,. (33 Usng relaton (2 n (33, we get C r r ( r (r!(m + r ( 2F (cγ r c, ( c ;. (34 Now n vew of (7 n (34, we have n vew of (22. g [r,n,m,k] (ydy C r r ( r ( k (r!(m + r ( B m + + (n r +,

6 Nayabuddn / ProbStat Forum, Volume 6, October 23, Pages Moment of Y [r,n,m,k] In vew of (29, we have ( E Y (a [r,n,m,k] y a g [r,n,m,k] (ydy C r (r!(m + (r!(m + p Lettng t + α 2 y n (35 we have r r c(c + ( r ( (cγ y a α r c, 2 ( + α 2 y (c+ 2 F (cγ r + 2 C r r r c(c + (cγ r c p ( p (cγ r + 2p! C r c(c + r ( r ( (α 2 a (r!(m + r cγ r + In vew of relaton (23, (36, becomes (α 2 a C r c(c + r ( r ( (r!(m + r cγ r + p ( ( r cγ r + ; α 2 y cγ r + dy y a (α 2 ( + α 2 y (c+ ( α 2y p dy. (35 p (cγ r c p ( p ( p (cγ r + 2p! (cγ r c p ( p ( p (cγ r + 2 p p! t (c+ ( t (p+a dt. (36 Γ(c a pγ(p + + a. (37 Γ(c + Now on usng ( n (37, we get after smplfcaton α a 2 C r ( r c(c + r ( (cγ Γ(c aγ( + a r c,, ( + a (r!(m + r 3F 2 cγ r + Γ(c + (cγ r + 2, ( c + a After applcaton of (2 n (38, we have cα2 a C r (c aγ(c a Γ( + a r ( r (r!(m + r ( 2F Γ(c + a, ( cγ r ( a Applyng (7 n (39, we get C r Γ(c + aγ( + a r ( r (α 2 a (r!(m + r ( cγ(c γ r a c, ;. (38. (39 α2 a C r Γ(c + aγ( + a r ( r ( k (r!(m + r ( a sb + (n r + Γ(c + m + c(m +,, whch after applcaton of (22, yelds ( E y (a [r,n,m,k] C r Γ(c + a Γ( + a Γ( k+(n r(m+ a c (α 2 a (m + r m+ cγ(c Γ( k+n(m+ a c Γ(c + a Γ( + a (α 2 a cγ(c r ( a cγ m+ (4 (4

7 Nayabuddn / ProbStat Forum, Volume 6, October 23, Pages Remark 3.. Set m, k n (4, to get moments of concomtants of order statstcs from bvarate Lomax dstrbuton ( E y (a [r:n] n! (α 2 a (n r! and at r, we get ( E y (a [:n] n (α 2 a (nc a Further, n vew of (, (42 becomes ( E y (a [r:n] n n r+ Γ(c + a Γ( + a Γ(n r + a c cγ(c Γ(n + a c Γ(c + a Γ( + a. (42 Γ(c ( ( n ( n+r Γ(c + a Γ( + a n r (c a(α 2 a. Γ(c Remark 3.2. At m n (4, we get moment of concomtants of kth upper record value from bvarate Lomax dstrbuton ( E y (a [r,n,,k] Γ(c + a Γ( + a (α 2 a cγ(c ( a. ck r Here n Table 3, t may be noted that the well known property of order statstcs n E(X :n ne(x (Davd and Nagaraa, 23 s satsfed. 4. Jont probablty densty functon of Y [r,n,m,k] and Y [s,n,m,k] For the bvarate Lomax dstrbuton as gven n (, usng (2, (3, (4 and (7 n (9, the ont pdf of rth and sth concomtants of gos Y [r,n,m,k] and Y [s,n,m,k] s gven as g [r,s,n,m,k] (y, y 2 C s c 2 (c + 2 (α (α 2 2 (r!(s r!(m + s 2 r s r ( + ( r ( s r ( + α x 2 c(γs ( + α x 2 + α 2 y 2 c+2 I(x 2, y dx 2 (43 where I(x 2, y x2 α ( + α x c(s r+ (m+ c ( + α x + α 2 y c+2 dx. (44 Let t ( + α x, then the R.H.S. of (44 reduces to I(x 2, y (λ β +αx 2 t α ( + t λ β dt, (45 where α c(s r + (m + c, β (c + 2 and λ α 2 y. Now on usng (2 n (45, and smplfyng, we get I(x 2, y (λ β p ( p (β p( λ p p! [ ] ( + α x 2 (α p. ( α + p +

8 Nayabuddn / ProbStat Forum, Volume 6, October 23, Pages Table : Mean of the concomtant of order statstcs n r α 2, c 2 α 2 2, c 3 α 2 3, c Table 2: Mean of the concomtant of record statstcs r α 2, c 2, k α 2 2, c 3, k 2 α 2 3, c 4, k

9 Therefore, n vew of (43, we have Nayabuddn / ProbStat Forum, Volume 6, October 23, Pages g [r,s,n,m,k] (y, y 2 c 2 (c + 2 (α (α 2 2 C s (r!(s r!(m + s 2 r s r ( + ( r ( s r (λ β p α ( + α x 2 (cγs c ( + α x 2 + α 2 y 2 c+2 [ ( + α x 2 (α p ( p (β p ( λ p ( α + p + p! ] dx. (46 Lettng t 2 ( + α x 2 n (46, and usng relaton (2, we get g [r,s,n,m,k] (y, y 2 c 2 (c + 2 (α 2 2 C s (r!(s r!(m + s 2 (λ β (δ β p where δ α 2 y 2 and θ cγ s c. Set d θ and g 2 θ α n (47, to get c2 (c + 2 (α 2 2 C s λ β δ β r (r!(s r!(m + s 2 s r (β p ( λ p p! After substtutng the value of λ and δ n (48, we get c 2 (c + 2 (α 2 2 C s (r!(s r!(m + s 2 l (β l ( l (d + ll! p r l s r (β l ( δ l l! ( ( ( + r s r r Usng relaton (3 and (4 n (49, t becomes Therefore c 2 (c + 2 C s (r!(s r!(m + s 2 (α 2 (α 2 y β (α 2 (α 2 y 2 β s r ( + ( r ( + ( r p ( s r ( θ + l(2 θ α + l + p, (47 (β p ( λ p (g + p + lp! ( s r l (β l ( δ l (d + ll!. (48 (α 2 y 2 β (α 2 y β (β p ( α 2y p (g + p + lp!. (49 l p r s r ( + ( r (g p+l (g + p+l (β l (d l (β p ( s r ( α 2y p (d + l p! ( l l! dg. g [r,s,n,m,k] (y, y 2 c 2 (c + 2 C s (r!(s r!(m + s 2 (α 2 (α 2 (α 2 y β (α 2 y 2 β F :; :2; r s r ( + ( r ( s r (g; (β; (d; (β; ;, α 2y (g + ; (d + ; dg (5 where F :; :2; [ ] s as defned n (25.

10 Nayabuddn / ProbStat Forum, Volume 6, October 23, Pages We now prove that g [r,s,n,m,k] (y, y 2 dy dy 2 A g [r,s,n,m,k] (y, y 2 dy dy 2. We have F :; :2; (α 2 (α 2 y β (α 2 (α 2 y 2 β (g; (β; (d; (β; ;, α 2y (g + ; (d + ; dy dy 2 where Then, A c 2 (c + 2 C s (r!(s r!(m + s 2 r g [r,s,n,m,k] (y, y 2 dy dy 2 A On applyng (5 n (5, we can wrte (α 2 A (α 2 y 2 β l s r (g l (β l (d l ( (g + l (d + l l! ( + ( r p l l { ( s r (α 2 (α 2 y β (α 2 (α 2 y 2 β (g l+p (g + l+p (β l (d l (β p (α 2 (α 2 y β dg. ( α 2y p (d + l p! (g + l p (β p ( (g + l + p p! p ( α 2y p l l! dy dy 2. (5 dy }dy 2. (52 Now usng relaton (26 n (52, we have α 2 A (α 2 y 2 β (g l (β l (d l ( (g + l (d + l l! l l (α 2 (α 2 y β 2 F (β; (g + l (g + + l ; α 2y dy dy 2. (53 Now lettng t α 2y (α 2 y 2 A (α 2 y 2 β n (53, to get (g l (β l (d l ( (g + l (d + l l! l l t (β 2F (β; (g + l (g + + l ; t dt dy 2. Now n vew of relaton (8, we have (α 2 A (α 2 y 2 β (g l (β l (d l ( (g + l (d + l l! l Further, usng relaton (3 n (54, to get ga (α 2 (g β + (α 2 y 2 β l (g β + l (β l (d l ( (g β + 2 l (d + l l! l (g + l (β (g + l + β dy 2. (54 l dy 2. (55 Now usng relaton (26 n (55, we have ga (g + β; (β; (d (α 2 (g β + (α 2 y 2 β 3 F 2 ; dy 2. (56 (g + 2 β; (d +

11 Set t 2 ga (g β + Nayabuddn / ProbStat Forum, Volume 6, October 23, Pages n (56, to get (t 2 (β 3F 2 Now on usng relaton (9 n (57, we have (g + β; (β; (d ; t 2 (g + 2 β; (d + dt 2. (57 Agd (β 2 (g + 2 2β(d + β. (58 Now after puttng the value of g, d, β, A n (58, we get Therefore, C s (r!(s r!(m + s n vew of (22. r s r g [r,s,n,m,k] (y, y 2 dy dy 2 s r ( + ( r ( s r [ k C s (r!(s r!(m + s ( ( + r ( k B m + + (n r +, B m+ + (n s + ] k [ m+ + (n r + ]. r ( s r ( k m + + (n s +,. 5. Product moments of two concomtants Y [r,n,m,k] and Y [s,n,m,k] The product moments of two concomtants Y [r,n,m,k] and Y [s,n,m,k] s gven by ( E Y (a [r,n,m,k] Y (b [s,n,m,k] In vew of (5 and (59, we have ( E Y (a [r,n,m,k] Y (b [s,n,m,k] A yy a 2 b (α 2 (α 2 (α 2 y β (α 2 y 2 β F :; :2; A On applyng (5 n (6, we have A y2 b (α 2 (α 2 y 2 β yy a 2 b (α 2 (α 2 (α 2 y β (α 2 y 2 β l y a y b 2 g [r,s,n,m,k] (y, y 2 dy dy 2. (59 (g l (β l (d l ( (g + l (d + l l! p l l (g; (β; (d; (β; ;, α 2y (g + ; (d + ; (g l+p (g + l+p (β l (d l (β p { Usng relaton (26 n (6, we have A y b (α 2 (g l (β l (d l ( l 2 (α 2 y 2 β (g + l (d + l l! l (β; (g + l y a (α 2 (α 2 y β 2 F (g + l + y a (α 2 (α 2 y β ( α 2y p (d + l p! ( l l! (β p (g + l p ( (g + + l p p! p ; α 2y α 2y p dy dy 2. dy dy 2. (6 dy }dy 2. (6 dy dy 2. (62

12 Settng t α 2y Nayabuddn / ProbStat Forum, Volume 6, October 23, Pages n (62, to get A (α 2 a y2 b (α 2 (g l (β l (d l ( l (α 2 y 2 β (g + l (d + l l! l (β; (g + l t (β a 2F ; t (g + l + dt dy 2. (63 On usng relaton (8 n (63, we have A (g + lγ(β a Γ(a + (α 2 a Γ(β(g + l + β + a y2 b (α 2 (α 2 y 2 β l (g l (β l (d l ( l (g + l (d + l l! dy 2. Now n vew of relaton (3 and (26, we have Let t 2 Γ(β a Γ(a + (g + β + a Γ(β (d; (g + β + a; (β 3 F 2 (d + ; (g + 2 β + a A (α 2 a g n (64, we have A g (α 2 a+b 3 F 2 (g + β + a Usng relaton (9 n (65, to get A Γ(β a Γ(a + (α 2 a+b Γ(β Γ(β a Γ(a + Γ(β y2 b ; (d; (g + β + a; (β ; t 2 (d + ; (g + 2 β + a Γ(β b Γ(b + Γ(β (α 2 (α 2 y 2 β dy 2. (64 t (β b 2 dt 2. (65 dg (d β + b + (g 2β + a + b + 2. (66 Now puttng the value of A, d, g and β n (66, we have ( E Y (a [r,n,m,k] Y (b c 2 (c + 2 C s Γ(c a + Γ(a + [s,n,m,k] (α 2 (a+b (r!(s r!(m + s 2 Γ(c + 2 Γ(c b + Γ(b + Γ(c + 2 r s r [c{k + (n s + (m + } b] ( + ( r ( s r [c{k + (n r + (m + } a b] (c + 2 C s Γ(c a + Γ(a + (α 2 (a+b (r!(s r!(m + s 2 Γ(c + 2 ( k r Γ(c b + Γ(b + Γ(c + 2 s r ( ( s r ( ( r B ( k b B + (n s + m + + (n r + a m + c(m +,. + b c(m +, (67

13 In vew of relaton (22, we get Nayabuddn / ProbStat Forum, Volume 6, October 23, Pages (c + 2 C s Γ(c a + Γ(a + Γ(c b + Γ(b + (α 2 (a+b (r!(s r!(m + s 2 Γ(c + 2 Γ(c + 2 ( k B + (n r a + b ( k m + c(m +, r b B + (n s m + c(m +, s r, whch after smplfcaton yelds ( E Y (a [r,n,m,k] Y (b C s [s,n,m,k] (α 2 (a+b Γ( k (m+ + (n r a+b Γ( k (m+ + n c(m+ a+b c(m+ Γ(c a + Γ(a + (m + s 2 Γ(c + k Γ( (m+ + (n s b + (n r b Γ( k (m+ c(m+ c(m+. Γ(c b + Γ(b + Γ(c + ( E Y (a [r,n,m,k] Y (b Γ(c a + Γ(a + Γ(c b + Γ(b + [s,n,m,k] (α 2 (a+b Γ(c + Γ(c + r a+b ( cγ s r+ ( b cγ. (68 Remark 5.. Set m, k n (67, to get product moments of concomtants of order statstcs from bvarate Lomax dstrbuton ( E Y (a [r:n] Y (b [s:n] C r,s:n r s r Γ(c a + Γ(a + Γ(c b + Γ(b + (α 2 (a+b Γ(c Γ(c ( ( r s r ( + [sc nc c c + b] [rc nc c c + a + b]. ( E Y (a [r:n] Y (b [s:n] (α 2 (a+b n! (n s! a+b Γ(n r + c b Γ(n s + c Γ(n + a+b c Γ(n r + b c. Γ(c a + Γ(a + Γ(c b + Γ(b + Γ(c + Γ(c + Remark 5.2. If m, n (68, then we get product moment of concomtants of kth upper record value from bvarate Lomax dstrbuton as ( E Y (a [r,n,,k] Y (b [s,n,,k] Γ(c a + Γ(a + (α 2 (a+b Γ(c + Γ(c b + Γ(b + Γ(c + ( a+b ck r ( b. ck (s r Acknowledgement Authors are grateful to Dr. Haseeb Athar, Algarh Muslm Unversty, Algarh, Inda, for hs help and suggestons throughout the preparaton of ths manuscrpt. The authors are also thankful to learned referee for hs/her comments whch led mprovement n the manuscrpt.

14 Nayabuddn / ProbStat Forum, Volume 6, October 23, Pages Table 3: Product moments between the concomtants of order statstcs: α 2 c 3 n s \ r Table 4: Product moments between the concomtants of order statstcs: α 2 2 c 4 n s \ r

15 Nayabuddn / ProbStat Forum, Volume 6, October 23, Pages Table 5: Product moments between the concomtants of order statstcs: α 2 3 c 5 n s \ r Table 6: Product moments between the concomtants of record statstcs: s r α 2, c 3 α 2 2, c 4 α 2 3, c

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