ON GENERALIZED SARMANOV BIVARIATE DISTRIBUTIONS

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1 TWMS J. A. Eng. Math. V., N., 2, ON GENERALIZED SARMANOV BIVARIATE DISTRIBUTIONS I. BAIRAMOV, B. (YAĞCI) ALTINSOY2, G. JAY KERNS 3 Abstract. A class of bivariate distributions which generalizes the Sarmanov class is introduced. This class ossesses a simle analytical form desirable deendence roerties. The admissible range for association arameter for given bivariate distributions are derived the range for correlation coefficients are also resented. Keywords: Sarmanov class of bivariate distributions, range for association arameter, correlation coefficient. AMS Subject Classification: 62H2, 62G3. Introduction Let (X, Y ) be a bivariate rom vector. Sarmanov (966) introduced a class of bivariate distributions for (X, Y ) having a joint robability density function (df) of the form h α (x, y) = f X (x)f Y (y) { + αψ (x)ψ 2 (y)}, () where f X (x) f Y (y) are the marginal df s of X Y, resectively, ψ (x) ψ 2 (y) are bounded nonconstant functions such that f X (t)ψ (t)d t =, f Y (t)ψ 2 (t)dt =. The association arameter α is a real number which satisfies the condition +αψ (x)ψ 2 (y) for all x y. In this aer, we deal with the concet of a coula. A two-dimensional coula is a function C(x, y) from [, ] [, ] to [, ] with the roerties:. C(x, ) = = C(, y), C(x, ) = x C(, y) = y; 2. For every x, x 2 y, y 2 such that x < x 2 y < y 2, we have C(x 2, y 2 ) C(x 2, y ) C(x, y 2 ) + C(x, y ). According to Sklar s Theorem, if F (x, y) is a joint distribution function with continuous marginals F X (x) F Y (y), then there exists a unique coula C such that F (x, y) = Izmir University of Economics, Deartment of Mathematics, 3533, Balcova, Izmir, Turkey, ismihan.bayramoglu@ieu.edu.tr 2 Ministry of Transort Communication of the Reublic of Turkey, Deartment of Strategy Develoment, Hakkı Turayliç Caddesi No:5 Emek, Ankara, Turkey, banuyagci@gmail.com 3 Youngstown University, Deartment of Mathematics Statistics, Youngstown, Ohio , USA, gkerns@ysu.edu Manuscrit received 5 May 2. 86

2 I. BAIRAMOV, G. JAY KERNS, B. (YAĞCI) ALTINSOY: ON GENERALIZED SARMANOV BIVARIATE...87 C(F X (x), F Y (y)). Theory alications of coulas are well documented in Nelsen (998). The coula generated by () has the form: C(x, y) = xy + x y ψ (t)ψ 2(s) d tds, where ψ (t) = ψ (F X (t)) ψ 2 (s) = ψ 2(F Y (s)). One may note in the secial case ψ (x) = 2F X (x) ψ 2 (y) = 2F Y (y) that the classical Farlie-Gumbel-Morgenstern (FGM) distributions are recovered. The range of the correlation coefficient for the FGM coula is /3 ρ /3. Although the FGM model is an interesting family constructed from secified marginals, this model cannot be used to reresent the joint distribution of two highly correlated variables. To increase deendence between variables, Huang Kotz (999) considered olynomial-tye single arameter extensions of FGM (with uniform marginals): C α (x, y) = xy { + α ( x ) ( y )},, x, y (2) C α (x, y) = xy { + α ( x) q ( y) q }, q, x, y. (3) The maximal ositive correlation for (2), namely ρ = 3/8, is attained for = 2, an imrovement over the case = for which ρ = /3. Bairamov Kotz (2) rovided several theorems characterizing symmetry deendence roerties of FGM distributions. They also roosed a modification by introducing additional arameters q: F,q,α (x, y) = xy { + α ( x ) q ( y ) q }, q >,, x, y. (4) For (4) the admissible range of α is { ( ) + q 2(q) min 2, } α (q ) ( ) + q q. (q ) The maximal minimal values of ρ are within the range { ( ) 2t 2 + q 2(q) (q, ) min 2, } ρ 2t 2 (q, ) q where t (x, y) = Γ(x+)Γ yγ 2 y x++ 2 y ( ) + q q, (5) q Γ(x) is the Gamma function. In this case, the maximal ositive correlation is ρ max =.52 attained at q =.496 = 3. Hence, the extension (4) can achieve correlation greater than /2 comared to the classical FGM where the correlation cannot be greater than /3. Bairamov et al. (2) considered a modification of the form F, 2,q,q 2,α (x, y) = xy [ + α ( x ) q ( y 2 ) q 2 ], (6), 2 ; q, q 2 > ; x, y which yielded a slight imrovement for the correlation coefficient over the case (4). They also derived recurrence relations between moments of concomitants of order statistics for this class of bivariate distributions.

3 88 TWMS J. APP. ENG. MATH. V., N., 2 Lee (996) considered, for the Sarmanov model, kernels of the tye ψ (x) = x µ X ψ 2 (y) = y µ Y, where µ X = E(X) µ Y = E(Y ). Lee showed that the range of the correlation coefficient for this family of distributions is determined by both the marginals as { max, µ X µ Y } { α min ( µ X )( µ Y ) µ X ( µ Y ), ( µ X )µ Y }. (7) For selected distributions, one can obtain from (7) the maximum ositive (negative) correlation. However, for uniform marginals, the range for the correlation coefficient is the same as the FGM coula, i.e. [/3, /3]. Bairamov et al. (2) considered kernels ψ (x) = ψ 2 (x) = x e nxq ( x) e n(x)q, where n,, q are ositive real numbers. In this case, the maximal correlation coefficient ρ max =.59 is achieved for n = 2, = 3, q = 2. Yue et al. (2) reviewed various bivariate models constructed from gamma marginals alied them to hydrological frequency analysis. They noticed that many bivariate models have mainly remained in the form of theoretical develoments seldom succeeded in gaining oularity among ractitioners in the field of hydrological frequency analysis. The main reason for this is that the mathematical exressions of some of these models are comlex therefore have comutational limitations. Recently, Lin Huang (2) consider a generalized version of Sarmanov family introduced earlier by Bairamov et al. (2) show that unlike the traditional Sarmanov the generalized one always has a correlation aroaching one regardless of the marginals, as long as the marginals are of the same tye. In this aer, we construct classes of bivariate distributions which are generalizations of Sarmanov Sarmanov-Lee models. These distributions have a simle analytical form like the FGM models, as in the normal case, the correlation coefficient ρ totally governs the deendence between the variables. Some deendence roerties of these distributions are also discussed. 2. The generalized Sarmanov coula Consider a class of bivariate distributions with absolutely continuous marginals F X, F Y given by the joint density f X,Y (x, y) = f X (x)f Y (y) { + αh[f (x), F Y (y)]}, (8) where f X (x), f Y (y) are secified marginal densities, H(x, y) is an integrable function defined on [, ] satisfying H(x, y) dx =, H(x, y)dy =, (9) α is a real number satisfying the condition that + αh(x, y) for all x, y. The corresonding bivariate distribution function has the form F X,Y (x, y) = F X (x)f Y (y) + α F X (x) F Y (y) H(t, s)dtds, < x, y <

4 I. BAIRAMOV, G. JAY KERNS, B. (YAĞCI) ALTINSOY: ON GENERALIZED SARMANOV BIVARIATE...89 the coula is C X,Y (x, y) = xy + α x y Define the ositive negative domains D + D by H(t, s) dtds, x, y. () D + = {(x, y) : x, y, H(x, y) > } D = {(x, y) : x, y, H(x, y) < }, resectively. The admissible range for α ensuring that (8) is a joint density can be exressed as max {H(x, y) (x, y) D + } α max {H(x, y) (x, y) D } max (x, y) D + (H(x, y)) α The correlation coefficient of X Y, if it exists, is given by ρ = α σ X σ Y F X. () max (H(x, y)) (x, y) D (x)f (y)h(x, y)dxdy, (2) where F X (x) = inf {y : F X(x) y} is the inverse cumulative distribution function for F X. Therefore in family (8), as in the normal case, the correlation coefficient ρ totally governs the deendence. Using the Cauchy-Schwarz inequality, we have ( ) /2 ( ) /2 /2 ρ α + µ2 X σx 2 + µ2 Y σy 2 H 2 (x, y) dxdy. The main justification for the interest in the family (8) is the constructive aroach involved in its definition the transarent manner in which the deendence is created. It can be shown that the family ossesses the TP 2 roerty, which is the strongest among the deendence roerties; see, for examle, Shaked Shantikumar (994). The air of variables (X, Y ) is said to be totally ositive deendent (TP 2 ) if its joint density f(x, y) satisfies the condition Y f(x, y )f(x 2, y 2 ) f(x 2, y )f(x, y 2 ) for all x < x 2 y < y 2. It is well known that for more details on deendence roerties see Barlow Proschan (98) Joe (99) the TP 2 roerty imlies the following imortant deendence roerties: () Stochastically increasing ositive deendence, SI(X Y ) : (X is stochastically increasing in Y if P {X > x Y = y} is a nondecreasing function of y for all x); (2) Right-tail increasing - RTI(X Y ) (X is right-tail increasing in Y if P {X > x Y > y} is increasing in y for each x);

5 9 TWMS J. APP. ENG. MATH. V., N., 2 (3) A(X, Y )association: (the rom vector (X, Y ) is associated if the inequality E [g (X, Y )g 2 (X, Y )] E [g (X, Y )] E [g 2 (X, Y )] holds for all real-valued functions g, g 2 which are increasing in each comonent such that the exectations exist); (4) Positive quadrant deendence - PQD (the rom vector (X, Y ) is ositive quadrant deendent if P {X x, Y y} P {X x} P {Y y}). For the oints x < x 2 y < y 2, denote A(x, x 2, y, y 2 ) = H(x 2, y 2 )H(x 2, y )H(x, y 2 )+H(x, y ) B(x, x 2, y, y 2 ) = H(x, y )H(x 2, y 2 ) H(x, y 2 ) + H(x 2, y ). Theorem 2.. The following holds: () For α, if A(x, x 2, y, y 2 ) B(x, x 2, y, y 2 ), then coula () satisfies the TP 2 roerty; (2) For α, if A(x, x 2, y, y 2 ) B(x, x 2, y, y 2 ), then coula () satisfies the TP 2 roerty. Proof. The assertion of the theorem is a direct consequence of the equalities f(x, y )f(x 2, y 2 ) f(x 2, y )f(x, y 2 ) = [ + αh(x, y )] [ + αh(x 2, y 2 )] [ + αh(x, y 2 )] [ + αh(x 2, y )] = αa(x, x 2, y, y 2 ) + α 2 B(x, x 2, y, y 2 ). Examle 2.. Consider the function H(x, y) = x + y ( + )x y, x, y, >. (3) + It is easy to verify that H(x, y) satisfies conditions (9). The corresonding bivariate density is of the form { f(x, y) = + α x + y ( + )x y }, x, y, >. (4) + Lemma 2.. For for α satisfying + α for < < for α satisfying + + 2( + ) ( + ) 2, α +, (3) reresents a joint robability density of two uniform rom variables. Proof. We will use the inequalities in (). It can be verified that H(x, y) x = x ( + )x y, H(x, y) y = y ( + )x y, H xx = 2 H(x, y) x 2 = ( )x 2 ( + )( )x 2 y,

6 I. BAIRAMOV, G. JAY KERNS, B. (YAĞCI) ALTINSOY: ON GENERALIZED SARMANOV BIVARIATE...9 H xy = 2 H(x, y) x y = ( + ) 2 x y, From the equations H(x,y) x H yy = 2 H(x, y) y 2 = ( )y 2 ( + )( )x y 2. = H(x,y) y be x = (+) / y = (+) /. Since =, we find the critical oints of H(x, y) to H xx (x, y )H yy (x, y ) [H xy (x, y )] 2 = <, ( + ) 2(2)/ (x, y ) is a saddle oint of H(x, y). It is also true that f(x, y) = f(x, y ) = f(x, y ) = for every x, y. The oints (, ) (, ) are local maxima of H(x, y) in D + H(, ) = H(, ) = The local minima of H(x, y) in D are (, ) (, ). For, For < <, H(, ) = 2 ( + )2 + ( + ) H(, ) = 2 ( + )2 + ( + ) 4 +. (5) < = H(, ). + (6a) The roof is comleted by using (), (5), (6a), (7). > = H(, ). (7) + The corresonding coula can be written as follows: C (x, y) = xy + α { x + y x + y + xy + xy +}, x, y. (8) + From (2), we then have ρ = 2α ( xy x + y ( + )x y ) + For, the range for the correlation coefficient is 3 ( + 2) 2 ρ 3 ( + 2) 2 ; for < <, the range for the correlation coefficient is 2 ( + 2) 2 ρ 3 ( + 2) 2. 3α 2 dxdy = ( + )( + 2) 2. The maximum ositive correlation is 3/8 =.375 which is achieved at the oint = 2. The maximum negative correlation is /3 it is achieved at the oint =. Corollary 2.. For α <, coula (8) has the TP 2 roerty.

7 92 TWMS J. APP. ENG. MATH. V., N., 2 Proof. For coula (8), B(x, x 2, y, y 2 ) = =. A(x, x 2, y, y 2 ) = ( + )(y 2 y )(x x 2 ) ( x + y ( + )x y + ( x + y 2 ( + )x y 2 + ) ( x 2 + y 2 ( + )x 2 y 2 + ) ( x 2 + y ( + )x 2 y + And since (y 2 y )(x x 2 ), we have A(x, x 2, y, y 2 ). Therefore, it follows from Theorem that if α <, (8) has the TP 2 roerty. Remark 2.. One may consider the kernel H (x, y) = H(x, y). Then the admissible range for the association arameter α is ( + ) 2( + ) ( + ) 2 α + for, ( + ) α + for < <. The corresonding coula is C 2 (x, y) = xy α { x + y x + y + xy + xy +}, x, y. (9) + The range for the correlation coefficient is 3 ( + 2) 2 ρ 3 ( + 2) 2 for 3 ( + 2) 2 ρ 2 for < <. ( + 2) 2 The maximum negative correlation is.375 the maximum ositive correlation is /3. Corollary 2.2. For α >, coula (9) has the TP 2 roerty. Examle 2.2. Consider the kernel H 2 (x, y) = x + y 2 x y, x, y. The corresonding bivariate df is f 2 (x, y) = + α { x + y 2 x y }, x, y. A similar analysis shows that the admissible range for α is α for < 2, (2) α for > 2. (2) ( ) 2 ) )

8 I. BAIRAMOV, G. JAY KERNS, B. (YAĞCI) ALTINSOY: ON GENERALIZED SARMANOV BIVARIATE...93 The corresonding coula is C 3 (x, y) = xy { + α [ x + y x y ]}, x, y. (22) The range for the correlation coefficient is 3( )2 3( ) ρ for < 2 (23) ( + ) 2 ( + ) 2 3 3( ) ρ for > 2. (24) ( + ) 2 ( + ) 2 Remark 2.2. As before, one may consider the kernel H 3 (x, y) = H 2 (x, y). The coula has the form C 4 (x, y) = xy { α [ x + y x y ]}, x, y. (25) Then the inequalities (2) (2) for α the inequalities (23) (24) for the correlation coefficient will be reversed. Corollary 2.3. For α <, coula (22) has the TP 2 roerty. Corollary 2.4. For α >, coula (25) has the TP 2 roerty. The roofs are similar to the roof of Corollary 2.. Examle 2.3. Now consider the following kernel which is different from those considered reviously: H 4 (x, y) = min(x, y) x y x2 y 2 +, x, y. (26) 2 It can be verified that the function (26) satisfies conditions (9). It is evident that max (x, y) D [H 4 (x, y)] = min (x, y) D H 4 (x, y) = H 4 (, ) = H 4 (, ) = 2, (27) maxh 4 (x, y) = H 4 (, ) =. (28) (x, y) D + The inequalities () along with equations (27) (28) give the admissible range for the association arameter α as α 2. From formula (2), the correlation coefficient is given by the quantity ρ = 2α { xy min(x, y) x y x2 y 2 + } 2 the range for the correlation coefficient is 9 4 ρ 9 2 dx dy = 9 4 α

9 94 TWMS J. APP. ENG. MATH. V., N., 2 (or.225 ρ.45). The corresonding coula has the form { xy C 5 (x, y) = xy + α 2 + x3 y 3 xy } {min(x, y)}3 max(x, y), { } xy + α xy 2 = + x3 y 3 6 xy2 2 x3 6, x < y { } xy + α xy 2 + x3 y 3 6 x2 y 2 y3 6, x y. Remark 2.3. Just as before, one may consider the kernel H 5 (x, y) = H 4 (x, y). Then the admissible range for the association arameter α is 2 α the range for the correlation coefficient is 9 2 ρ 9 4. The simle analytical form of the resented coulas is sufficiently robust to allow alications in many ractical roblems. But even more flexibility may be added by introducing additional arameters to model (26). 3. Multivariate extension One can extend the generalized Sarmanov family of coulas to the multivariate case. Let H k (x, x 2,..., x k ), k = 2, 3,..., n, be a family of symmetric functions which have the following recurrence roerties: H k (x i, x i2,..., x ik, x ik ) x ik = H k (x i, x i2,..., x ik ), (29) (n)... H n (x i, x i2,..., x in ) dx i dx i2 dx in =, (3) for any k = 2, 3,..., n; i, i 2,..., i k {, 2,..., k} x i, x i2,..., x ik. Define now the following nvariate df with marginal distribution functions F i (x i ) marginal dfs f i (x i ): f α (x i, x i2,..., x in ) = f i (x i )f i2 (x i2 )...f in (x in ) { + α n H n (F i (x i2 ), F i2 (x i2 ),..., F in (x in ))}, (3) where the association arameter α n satisfies the condition + α n H n (x, x 2,..., x n ) for any x, x 2,..., x n. Denote D + n = {(x, x 2,..., x n ) : H n (x, x 2,..., x n ) } D n = {(x, x 2,..., x n ) : H n (x, x 2,..., x n ) }. Then, as in the bivariate case, the function given in (3) is a df of nvariate rom vector (X, X 2,..., X n ) if max H n (x, x 2,..., x n ) α n max (H n (x, x 2,..., x n )). (x, x 2,..., x n ) D n + (x, x 2,..., x n ) Dn (32)

10 I. BAIRAMOV, G. JAY KERNS, B. (YAĞCI) ALTINSOY: ON GENERALIZED SARMANOV BIVARIATE...95 For uniform marginals, we have c α (x, x 2,..., x n ) = + α n H n (x, x 2,..., x n ), x, x 2,..., x n. The corresonding nvariate coula is C(x, x 2,..., x n ) = x x 2...x n + α n Consider for examle a kernel x n x n x x, x 2,..., x n. H n (t, t 2,..., t n ) dt dt 2 dt n, H n (x, x 2,..., x n ) = x + x x n ( + ) n x x 2 x n n +. (33) It can be checked easily that the kernel satisfies conditions (29) (3). Therefore, one can consider the n variate df f n (x, x 2,..., x n ) = f (x )f 2 (x 2 ) f n (x n ) { + α n [F (x ) + F 2 (x 2) + + Fn(x n ) ( + ) n (F (x )F 2 (x 2 ) F n (x n )) n ]}, + < x, x 2,..., x n <. For uniform (, ) marginals, we have { c α (x, x 2,..., x n ) = + α n x + x x n ( + ) n x x 2 x n n } + the n variate coula is C α (x, x 2,..., x n ) = x x 2 x n { + α n + [ n n ]} x i x i (n ). (34) Theorem 3.. The admissible range for α n allowing (34) to be nvariate coula is + (n ) α n i= + ( + ) n n( + ) + (n ) i= + (n ) α n + n for < < n n. for n n, Proof. We will use (32). It can be seen that in Dn the function (33) takes its minimum value at the oint x = x 2 =... = x n =, i.e., H n (,,..., ) = n ( + ) n n +. (35) In fact, if at least one of the coordinates is equal to the others are equal to, then the second term in (33) is equal to. In this case the minimum value is obtained when one of the x i s is equal to, all others are equal to, i.e., H n (,,...,, ) = n +. (36)

11 96 TWMS J. APP. ENG. MATH. V., N., 2 Comare (35) with (36). If n(+) n <, i.e. > (n) n, then Hn (,,..., ) < H n (,,...,, ). Now when all of x i s are equal to zero, then H n (,,..., ) = n +. (37) Comare (37) with (35). If n ( + ) n <, i.e. > n n, then Hn (,,..., ) < H n (,,..., ). Therefore, if > n n, then + α n ( + ) n n( + ) + (n ). Now in D n + the function (33) has its maximum when one of the x i s is equal to all others are equal to. This value is H(,,...,, ) = (n). Therefore, if > n n, then + (n ) α n + + ( + ) n n( + ) + (n ). References [] Barlow R. Proschan, F., (98), Statistical Theory of Reliability Life Testing: Probability Models, to begin with, Silver Sring, MD. [2] Bairamov, I.G., Kotz, S. Bekçi, M., (2), New generalized Farlie-Gumbel-Morgenstern distributions concomitants of order statistics, Journal of Alied Statistics, 28(5), [3] Bairamov, I.G. Kotz, S. (22) Deendence structure symmetry of Huang-Kotz FGM distributions their extensions. Metrika. 56,, [4] Bairamov, I. G., Kotz, S. Gebizlioglu O.L. (2) The Sarmanov family its generalization. South African Statistical Journal. 35, [5] Huang, J.S. Kotz, S., (999), Modifications of the Farlie-Gumbel-Morgenstern distributions, A tough hill to climb, Metrika, 49, [6] Joe H., (997), Multivariate Models Deendence Concets, Chaman Hall, London. [7] Lee, M.-L. T., (996), Proerties alications of the Sarmanov family of bivariate distributions, Communications in Statistics. -Theory Meth., 25(6), [8] Lin, G.D. Huang, J.S., (2), Maximum correlation for the generalized Sarmanov bivariate distributions, Journal of Statistical Planning Inference, 4, [9] Nelsen, R.B., (998), An Introduction to Coulas, Sringer-Verlag, New York. [] Sarmanov, O.V., (966), Generalized normal correlation two-dimensional Frechet classes, Doklady (Sovyet Mathematics), Tom 68, [] Shaked, M., Shanthikumar, J.G., (994), Stochastic Orders Their Alications, Academic Press. Boston. [2] Yu, S., Ouarda, T.B.M.J. Bobée, B., (2), A rewiew of bivariate gamma distributions for hydrological alication, Journal of Hydrology, 246, -8.

12 I. BAIRAMOV, G. JAY KERNS, B. (YAĞCI) ALTINSOY: ON GENERALIZED SARMANOV BIVARIATE...97 Ismihan Bairamov is resently Professor of Mathematics Statistics Dean of Faculty of Arts Sciences, Izmir University of Economics, Turkey. He graduated from Faculty of Alied Mathematics of Azerbaijan State University received his PhD degree in robability statistics from Kiev University, Ukraine in 988. He worked in Institute of Cybernetics of Academy of Sciences of Azerbaijan, Azerbaijan State University Azerbaijan Aerosace Agency as a researcher lecturer. Between he worked as a rofessor teaching robability statistic courses in Ankara University from 2 he teaches in deartment of Mathematics of Izmir University of Economics. His research interests focus mainly in theory of statistics, robability, theory order statistics, nonarametric statistics multivariate distributions coulas. Banu (Yağcı) Altınsoy is resently Statistician at Ministry of Transort Communication, Deartment of Strategy Develoment. She graduated from Faculty of Science, Deartment of Statistics, Ankara University received her PhD degree in Statistics from Ankara University, Turkey in 29. She worked in Deartment of Statistics, Faculty of Science, Ankara University as assistant between In June, 29, she started to work at Ministry of Transort Communication, Reublic of Turkey. Her research interests focus mainly in theory order statistics, nonarametric statistics, coulas, alied statistics environmental statistics. She is the member of the environmental studies at the ministry also rearing annual Turkey National Inventory Reort of UNFCCC (United Nations Framework Convention on Climate Change). G. Jay Kerns is resently Associate Professor at Deartment of Mathematics Statistics Youngstown University, USA. He has B.A.Ed. Degree in Mathematics from Glenville State College (999), M.A. Ph.D. degrees in Statistics from Bowling Green State University (2, 24). His research interests include Statistical Comuting, Exchangeability in Statistics, Exchangeable Measures the Primary Teaching Areas are Statistics Probability. He has ublished research aers dealing with De Finetti s Theorem for Abstract Finite Exchangeable Sequences, Calibration Exeriments in Engineering Quality Control, Alications of Statistics in Medical Systems Ecology.