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International Journal of Mathematical Analysis Vol. 10, 2016, no. 10, 455 - 467 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2016.6219 A Bivariate Distribution whose Marginal Laws are Gamma and Macdonald Daya K. Nagar, Edwin Zarrazola and Luz Estela Sánchez Instituto de Matematicas Universidad de Antioquia Calle 67, No. 53108, Medellin, Colombia c 2016 Daya K. Nagar, Edwin Zarrazola and Luz Estela Sánchez. This article Copyright is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Gamma and Maconald distributions are associated with gamma and extended gamma functions, respectively. In this article, we define a bivariate distribution whose marginal distributions are gamma and Macdonald. We study several properties of this distribution. Mathematics Subject Classification: 33E99, 60E05 Keywords: Confluent hypergeometric function; entropy; extended gamma function; gamma distribution; Laguerre polynomial 1 Introduction The gamma function was first introduced by Leonard Euler in 1729 as the limit of a discrete expression and later as an absolutely convergent improper integral, Z ∞ Γ(a) = ta−1 exp(−t) dt, Re(a) > 0. (1) 0 The gamma function has many beautiful properties and has been used in almost all the branches of science and engineering. Replacing t by z/σ, σ > 0, 456 Daya K. Nagar, Edwin Zarrazola and Luz Estela Sánchez in (1), a more general definition of gamma function can be given as Z ∞ z −a z a−1 exp − dz, Re(a) > 0. Γ(a) = σ σ 0 (2) In statistical distribution theory, gamma function has been used extensively. Using integrand of the gamma function (2), the gamma distribution has been defined by the probability density function (p.d.f.) v a−1 exp(−v/σ) , σ a Γ(a) a > 0, σ > 0, v > 0. (3) We will write V ∼ G(a, σ) if the density of V is given by (3). Here, a and σ determine the shape and scale of the distribution. In 1994, Chaudhry and Zubair [3] defined the extended gamma function, Γ(a; σ), as Z ∞ σ dt, Γ(a; σ) = ta−1 exp −t − t 0 where σ > 0 and a is any complex number. For Re(a) > 0 and by taking σ = 0, it is clear that the above extension of the gamma function reduces to the classical gamma function, Γ(a, 0) = Γ(a). The generalized gamma function (extended) has been proved very useful in various problems in engineering and physics, see for example, Chaudhry and Zubair [2–6]. Using the integrand of the extended gamma function, an extended gamma distribution can be defined by the p.d.f. v a−1 exp (−v − σ/v) , Γ(a; σ) v > 0. The distribution given by the above density will be designated as EG(a, σ). By using the definition of the extended gamma function, Chaudhry and Zubair [4] have introduced a one parameter Macdonald distribution. By making a slight change in the density proposed by Chaudhry and Zubair [4], a three parameter Macdonald distribution (Nagar, Roldán-Correa and Gupta [7, 8]) is defined by the p.d.f. fM (y; α, β, σ) = σ −β y β−1 Γ(α; σ −1 y) , Γ(β)Γ(α + β) y > 0, σ > 0, β > 0, α + β > 0. We will denote it by Y ∼ M (α, β, σ). If σ = 1 in the density above, then we will simply write Y ∼ M (α, β). By replacing Γ(α; σ −1 y) by its integral representation, the three parameter Macdonald density can also be written as Z ∞ y α−1 σ −β y β−1 fM (y; α, β, σ) = exp −x − x dx, y > 0, (4) Γ(β)Γ(α + β) 0 σx A bivariate distribution whose marginal laws are gamma and Macdonald 457 where σ > 0, β > 0 and α + β > 0. Now, consider two random variables X and Y such that the conditional distribution of Y given X is gamma with the shape parameter β and the scale parameter σx and the marginal distribution of X is a standard gamma with the shape parameter α + β. That is f (y|x) = y β−1 exp(−y/σx) , Γ(β)(σx)β y>0 and g(x) = xα+β−1 exp(−x) , Γ(α + β) x > 0. Then (4) can be written as Z ∞ f (y|x)g(x) dx. fM (y; α, β, σ) = 0 Thus, the product f (y|x)g(x) can be used to create a bivariate density with Macdonald and standard gamma as marginal densities of Y and X, respectively. We, therefore, define the bivariate density of X and Y as f (x, y; α, β, σ) = xα−1 y β−1 exp (−x − y/σx) , σ β Γ(β)Γ(α + β) x > 0, y > 0, (5) where β > 0, α + β > 0 and σ > 0. The distribution defined by the density (5) may be called the Macdonald-gamma distribution. The bivariate distribution defined by the above density has many interesting features. For example, the marginal and the conditional distributions of Y are Macdonald and gamma, the marginal distribution of X is gamma, and the conditional distribution of X given Y is extended gamma. The gamma distribution has been used to model amounts of daily rainfall (Aksoy [1]). In neuroscience, the gamma distribution is often used to describe the distribution of inter-spike intervals (Robson and Troy [9]). The gamma distribution is widely used as a conjugate prior in Bayesian statistics. It is the conjugate prior for the precision (i.e. inverse of the variance) of a normal distribution. Further, the fact that marginal distributions are gamma makes this bivariate distribution a potential candidate for many real life problems. In this article, we study distributions defined by the density (5), derive properties such as marginal and conditional distributions, moments, entropies, information matrix, and distributions of sum and quotient. 458 2 Daya K. Nagar, Edwin Zarrazola and Luz Estela Sánchez Properties Let us now briefly discuss the shape of (5). The first order derivatives of ln f (x, y; α, β, σ) with respect to x and y are fx (x, y) = α−1 y ∂ ln f (x, y; α, β, σ) = + 2 −1 ∂x x σx (6) β−1 1 ∂ ln f (x, y; α, β, σ) = − ∂y y σx (7) and fy (x, y) = respectively. Setting (6) and (7) to zero, we note that (a, b), a = α + β − 2, b = σ(β − 1)(α + β − 2) is a stationary point of (5). Computing second order derivatives of ln f (x, y; α, β, σ), from (6) and (7), we get fxx (x, y) = α−1 2y ∂ 2 ln f (x, y; α, β, σ) = − 2 − 3, 2 ∂x x σx (8) ∂ 2 ln f (x, y; α, β, σ) 1 = , ∂x∂y σx2 (9) ∂ 2 ln f (x, y; α, β, σ) β−1 =− 2 . 2 ∂y y (10) fxy (x, y) = and fyy (x, y) = Further, from (8), (9) and (10), we get fxx (a, b) = − α + 2β − 3 , (α + β − 2)2 fyy (a, b) = − σ 2 (β 1 − 1)(α + β − 2)2 (11) and fxx (a, b)fyy (a, b) − [fxy (a, b)]2 = σ 2 (β 1 . − 1)(α + β − 2)3 (12) Now, observe that • If β > 1 and α+β > 2, then fxx (a, b)fyy (a, b)−[fxy (a, b)]2 > 0, fxx (a, b) < 0 and fyy (a, b) < 0 and therefore (a, b) is a maximum point. • If β < 1 and α+β < 2, then fxx (a, b)fyy (a, b)−[fxy (a, b)]2 > 0, fxx (a, b) > 0 and fyy (a, b) > 0 and therefore (a, b) is a minimum point. • If β < 1 and α + β > 2, then fxx (a, b)fyy (a, b) − [fxy (a, b)]2 < 0, and therefore (a, b) is a saddle point. A bivariate distribution whose marginal laws are gamma and Macdonald 459 A distribution is said to be positively likelihood ratio dependent (PLRD) if the density f (x, y) satisfies f (x1 , y1 )f (x2 , y2 ) ≥ f (x1 , y2 )f (x2 , y1 ) (13) for all x1 > x2 and y1 > y2 . In the present case (13) is equivalent to y1 x2 + x1 y2 ≤ x1 y1 + x2 y2 which clearly holds. Olkin and Liu [11] have listed a number of properties of PLRD distributions. By definition, the product moments of X and Y associated with (5) are given by Z ∞ Z ∞ α+r−1 β+s−1 x y exp (−x − y/σx) r s dy dx E(X Y ) = β σ Γ(β)Γ(α + β) 0 0 Z ∞ σ s Γ(β + s) = xα+β+r+s−1 exp (−x) dx Γ(β)Γ(α + β) 0 σ s Γ(β + s)Γ(α + β + r + s) , (14) = Γ(β)Γ(α + β) where both the lines have been derived by using the definition of gamma function. For r = −s, the above expression reduces to σ s Γ(β + s) E(X Y ) = , Γ(β) −s s (15) which shows that Y /σX has a standard gamma distribution with shape parameter β. Substituting appropriately in (14), means and variances of X and Y and the covariance between X and Y are computed as E(X) = α + β, E(Y ) = σβ(α + β), Var(X) = α + β, Var(Y ) = σ 2 β(α + β)(α + 2β + 1), and Cov(X, Y ) = σβ(α + β). The correlation coefficient between X and Y is given by s β ρX,Y = . α + 2β + 1 The variance-covariance matrix Σ of the random vector (X, Y ) whose bivariate density is defined by (5) is given by 1 σβ Σ = (α + β) σβ σ 2 β(α + 2β + 1) 460 Daya K. Nagar, Edwin Zarrazola and Luz Estela Sánchez Further, the inverse of the covariance matrix is given by 1 σ(α + 2β + 1) −1 −1 . Σ = 1 −1 σ(α + β)(α + β + 1) βσ The well-known Mahalanobis distance is given by 1 Y2 2 2 D = σ(α + 2β + 1)X − 2XY + σ(α + β)(α + β + 1) βσ − 2σ(α + β)(α + β + 1)X + σ(α + β)2 (α + β + 1) with E(D2 ) = 2 and E(D4 ) = 2 β(α + β)(α + β + 1) × β(α + β)(α + β + 4) + 3(β + 1)(α + β + 2)(α + β + 3) From the construction of the bivariate density (5), it is cleat that Y ∼ M (α, β, σ), X ∼ G(α + β), Y |x ∼ G(β, σx) and X|y ∼ EG(α, y/σ). Making the transformation S = X + Y and R = Y /(X + Y ) with the Jacobian G(y, x → r, s) = s in (5), the joint density of R and S is given by (1 − r)α−1 rβ−1 sα+β−1 exp [−s + rs − r/σ(1 − r)] fR,S (r, s; α, β, σ) = , σ β Γ(β)Γ(α + β) where 0 < r < 1 and s > 0. Now, integrating s by using gamma integral, the marginal density of R is derived as fR (r; α, β, σ) = (1 − r)−β−1 rβ−1 exp [−r/σ(1 − r)] , σ β Γ(β) 0 < r < 1. From the above density it can easily be shown that R/σ(1 − R) = Y /σX has a standard gamma distribution with shape parameter β. By integrating r, the marginal density of S is derived as Z sα+β−1 exp (−s) 1 r α−1 β−1 fS (s; α, β, σ) = β (1 − r) r exp rs − dr. σ Γ(β)Γ(α + β) 0 σ(1 − r) Now, writing exp − ∞ X r 1 m = (1 − r) r Lm , σ(1 − r) σ m=0 A bivariate distribution whose marginal laws are gamma and Macdonald 461 where Lm (x) is the Laguerre polynomial of degree m, and integrating r by using the integral representation of confluent hypergeometric function, we get the marginal density of S as Z 1 ∞ 1 sα+β−1 exp (−s) X (1 − r)α+1−1 rβ+m−1 exp (rs) dr Lm fS (s; α, β, σ) = β σ Γ(β)Γ(α + β) m=0 σ 0 ∞ Γ(α + 1)sα+β−1 exp (−s) X 1 Γ(β + m) = L m σ β Γ(β)Γ(α + β) σ Γ(α + β + m + 1) m=0 × 1 F1 (β + m; α + β + m + 1; s), s > 0, where α + 1 > 0, β > 0 and σ > 0. 3 Central Moments By definition, the (i, j)-th central joint moment of (X, Y ) is given by µij = E[(X − µX )i (Y − µY )j ]. For different values of i and j, expressions for µij are given by µ30 µ21 µ12 µ03 µ40 µ31 µ22 µ13 µ04 µ50 µ41 µ32 4 = 2(α + β), = 2σβ(α + β), = 2σ 2 β(α + β)(α + 2β + 1), = 2σ 3 β(α + β)[(α + 2β + 1)2 + (β + 1)(α + β + 1)], = 3(α + β)(α + β + 2), = 3σβ(α + β)(α + β + 2), = σ 2 β(α + β)[3β + (α + β + 1)(α + 4β + 6)], = 3σ 3 β(α + β)[2(β + 1)(α + β + 1)(α + 2β + 2) − β(α + β)(α + 2β + 1)], = 3σ 4 β(α + β)[2(β + 1)(α + β + 1)(α + 2β + 2)(α + 2β + 3) − β(α + β)(α + 2β + 1)2 ], = 4(α + β)(5α + 5β + 6), = 4σβ(α + β)(5α + 5β + 6), = 4σ 2 β(α + β)[4α + 6 + (α + β + 2)(2α + 7β)]. Entropies In this section, exact forms of Rényi and Shannon entropies are determined for the bivariate distribution defined in Section 1. 462 Daya K. Nagar, Edwin Zarrazola and Luz Estela Sánchez Let (X , B, P) be a probability space. Consider a p.d.f. f associated with P, dominated by σ−finite measure µ on X . Denote by HSH (f ) the well-known Shannon entropy introduced in Shannon [13]. It is define by Z f (x) ln f (x) dµ. (16) HSH (f ) = − X One of the main extensions of the Shannon entropy was defined by Rényi [12]. This generalized entropy measure is given by HR (η, f ) = ln G(η) 1−η (for η > 0 and η 6= 1), (17) where Z G(η) = f η dµ. X The additional parameter η is used to describe complex behavior in probability models and the associated process under study. Rényi entropy is monotonically decreasing in η, while Shannon entropy (16) is obtained from (17) for η ↑ 1. For details see Nadarajah and Zografos [10], Zografos and Nadarajah [15] and Zografos [14]. Now, we derive the Rényi and the Shannon entropies for the bivariate density defined in (5). Theorem 4.1. For the bivariate distribution defined by the p.d.f. (5), the Rényi and the Shannon entropies are given by HR (η, f ) = 1 [ln Γ[η(β − 1) + 1] + ln Γ[η(α + β − 2) + 2] − (η − 1) ln σ 1−η − [η(α + 2β − 3) + 3] ln η − η ln Γ(β) − η ln Γ(α + β)] (18) and HSH (f ) = −[(β − 1)ψ(β) + (α + β − 2)ψ(α + β) − ln σ − (α + 2β) − ln Γ(β) − ln Γ(α + β)]. (19) Proof. For η > 0 and η 6= 1, using the p.d.f. of (X, Y ) given by (5), we have Z ∞Z ∞ G(η) = [f (x, y; α, β, σ)]η dy dx 0 0 Z ∞Z ∞ 1 ηy η(α−1) η(β−1) = β x y exp −ηx − dy dx [σ Γ(β)Γ(α + β)]η 0 σx 0 Z ∞ Γ[η(β − 1) + 1] xη(α+β−2)+1 exp (−ηx) dx = η(2β−1)+1 η(β−1)+1 η σ η [Γ(β)Γ(α + β)] 0 Γ[η(β − 1) + 1]Γ[η(α + β − 2) + 2] = η−1 η(α+2β−3)+3 , σ η [Γ(β)Γ(α + β)]η A bivariate distribution whose marginal laws are gamma and Macdonald 463 where, to evaluate above integrals, we have used the definition of gamma function. Now, taking logarithm of G(η) and using (17), we get (18). The Shannon entropy (19) is obtained from (18) by taking η ↑ 1 and using L’Hopital’s rule. 5 Fisher Information Matrix In this section we calculate the Fisher information matrix for the bivariate distribution defined by the density (5). The information matrix plays a significant role in statistical inference in connection with estimation, sufficiency and properties of variances of estimators. For a given observation vector (x, y), the Fisher information matrix for the bivariate distribution defined by the density (5) is defined as 2 2 2 ∂ ln L(α,β,σ) ∂ ln L(α,β,σ) L(α,β,σ) E E E ∂ ln ∂α 2 ∂ β∂α ∂σ ∂α 2 2 2 L(α,β,σ) E ∂ ln ∂β E ∂ ln∂βL(α,β,σ) − E ∂ ln∂βL(α,β,σ) , 2 ∂α ∂σ 2 2 2 L(α,β,σ) L(α,β,σ) E ∂ ln∂σL(α,β,σ) E ∂ ln∂σ∂ E ∂ ln ∂σ 2 ∂α β where L(α, β, σ) = ln f (x, y; α, β, σ). L(α, β, σ) is obtained as From (5), the natural logarithm of ln L(α, β, σ) = −β ln σ − ln Γ(β) − ln Γ(α + β) + (α − 1) ln x y , + (β − 1) ln y − x − σx where x > 0 and y > 0. The second order partial derivatives of ln L(α, β, σ) are given by ∂ 2 ln L(α, β, σ) = −ψ1 (α + β), ∂α2 ∂ 2 ln L(α, β, σ) = −ψ1 (β) − ψ1 (α + β), ∂β 2 ∂ 2 ln L(α, β, σ) β 2y = 2− 3 , 2 ∂σ σ σ x 2 ∂ ln L(α, β, σ) = −ψ1 (α + β), ∂α ∂β ∂ 2 ln L(α, β, σ) = 0, ∂α ∂σ ∂ 2 ln L(α, β, σ) 1 =− , ∂β ∂σ σ 464 Daya K. Nagar, Edwin Zarrazola and Luz Estela Sánchez where ψ1 (·) is the trigamma function. Now, noting that the expected value of a constant is the constant itself and Y /σX follows a standard gamma distribution with shape parameter β, we have ∂ 2 ln L(α, β, σ) E ∂α2 ∂ 2 ln L(α, β, σ) E ∂β 2 = −ψ1 (α + β), = −ψ1 (β) − ψ1 (α + β), ∂ 2 ln L(α, β, σ) E ∂σ 2 ∂ 2 ln L(α, β, σ) E ∂α ∂β =− = −ψ1 (α + β), ∂ 2 ln L(α, β, σ) E ∂α ∂σ ∂ 2 ln L(α, β, σ) E ∂β ∂σ 6 β , σ2 = 0, 1 =− . σ Estimation The density given by (5) is parameterized by (α, β, σ). Here, we consider estimation of these three parameters by the method of maximum likelihood. Suppose (x1 , y1 ), . . . , (xn , yn ) is a random sample from (5). The loglikelihood can be expressed as: ln L(α, β, σ) = −nβ ln σ − n ln Γ(β) − n ln Γ(α + β) + (α − 1) n n X X yi . + (β − 1) ln yi − xi + σx i i=1 i=1 n X ln xi i=1 The first-order derivatives of this with respect to the three parameters are: n X ∂ ln L(α, β, σ) = −nψ(α + β) + ln xi , ∂α i=1 A bivariate distribution whose marginal laws are gamma and Macdonald 465 n X ∂ ln L(α, β, σ) = −n ln σ − nψ(β) − nψ(α + β) + ln yi , ∂β i=1 and n ∂ ln L(α, β, σ) nβ 1 X yi =− + 2 . ∂σ σ σ i=1 xi The maximum likelihood estimators of (α, β, σ), say (α̂, β̂, σ̂), are the simultaneous solutions of the above three equations. It is straightforward to see that β̂ can be calculated by solving numerically the equation " n !# n X yi yi 1 X ln − ln . ψ(β̂) − ln β̂ = n i=1 xi x i i=1 Further, for β̂ so obtained, σ̂ and α̂ can be computed by solving numerically the equations n 1 X yi β̂ σ̂ = n i=1 xi and n 1X ψ(α̂ + β̂) = ln xi n i=1 respectively. Using the expansion of the digamma function, namely, 1 1 23 17 10099 1 − + − − + ··· ψ(z) = ln z + + 2 24 z 48 z 2 5760 z 3 3840 z 4 2903040 z 5 an approximation for β̂ can be given as −1 1 q̃ β̂ = −1 , 2 (nq̄)1/n P Q 1/n and qi = yi /xi , i = 1, . . . , n. Using this where q̄ = ni=1 qi /n, q̃ = ( ni=1 qi ) estimate of β, the estimates of σ and α are given by q̃ σ̂ = 2q̄ −1 (nq̄)1/n and −1 1 (nq̄)1/n α̂ = x̃ − 1− , 2 q̃ Q 1/n respectively, where x̃ = ( ni=1 xi ) . Acknowledgements. The research work of DKN and LES was supported by the Sistema Universitario de Investigación, Universidad de Antioquia under the project no. IN10231CE. 466 Daya K. Nagar, Edwin Zarrazola and Luz Estela Sánchez References [1] H. Aksoy, Use of gamma distribution in hydrological analysis, Turkish Journal of Engineering and Environmental Sciences, 24 (2000), 419–428. [2] M. Aslam Chaudhry and Syed M. Zubair, On the decomposition of generalized incomplete gamma functions with applications to Fourier transforms, Journal of Computational and Applied Mathematics, 59 (1995), 253–284. http://dx.doi.org/10.1016/0377-0427(94)00026-w [3] M. A. Chaudhry and S. M. Zubair, Generalized incomplete gamma functions with applications, Journal of Computational and Applied Mathematics, 55 (1994), 99–123. http://dx.doi.org/10.1016/0377-0427(94)90187-2 [4] M. Aslam Chaudhry and Syed M. Zubair, Extended gamma and digamma functions, Fractional Calculus and Applied Analysis, 4 (2001), no. 3, 303– 324. [5] M. Aslam Chaudhry and Syed M. Zubair, Extended incomplete gamma functions with applications, Journal of Mathematical Analysis and Applications, 274 (2002), no. 2, 725–745. http://dx.doi.org/10.1016/s0022-247x(02)00354-2 [6] M. Aslam Chaudhry and Syed M. Zubair, On an extension of generalized incomplete Gamma functions with applications, Journal of the Australian Mathematical Society (Series B)- Applied Mathematics, 37 (1996), no. 3, 392–405. http://dx.doi.org/10.1017/s0334270000010730 [7] Daya K. Nagar, Alejandro Roldán-Correa and Arjun K. Gupta, Extended matrix variate gamma and beta functions, Journal of Multivariate Analysis, 122 (2013), 53–69. http://dx.doi.org/10.1016/j.jmva.2013.07.001 [8] Daya K. Nagar, Alejandro Roldán-Correa and Arjun K. Gupta, Matrix variate Macdonald distribution, Communications in Statistics - Theory and Methods, 45 (2016), no. 5, 1311–1328. http://dx.doi.org/10.1080/03610926.2013.861494 [9] J. G. Robson and J. B. Troy, Nature of the maintained discharge of Q, X, and Y retinal ganglion cells of the cat, Journal of the Optical Society of America A, 4 (1987), 2301–2307. http://dx.doi.org/10.1364/josaa.4.002301 [10] S. Nadarajah and K. Zografos, Expressions for Rényi and Shannon entropies for bivariate distributions, Information Sciences, 170 (2005), no. 2-4, 173–189. http://dx.doi.org/10.1016/j.ins.2004.02.020 A bivariate distribution whose marginal laws are gamma and Macdonald 467 [11] Ingram Olkin and Ruixue Liu, A bivariate beta distribution, Statistics & Probability Letters, 62 (2003), no. 4, 407–412. http://dx.doi.org/10.1016/s0167-7152(03)00048-8 [12] A. Rényi, On measures of entropy and information, Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. I, Univ. California Press, Berkeley, Calif., (1961), 547–561. [13] C. E. Shannon, A mathematical theory of communication, Bell System Technical Journal , 27 (1948), 379–423, 623–656. http://dx.doi.org/10.1002/j.1538-7305.1948.tb01338.x http://dx.doi.org/10.1002/j.1538-7305.1948.tb00917.x [14] K. Zografos, On maximum entropy characterization of Pearson’s type II and VII multivariate distributions, Journal of Multivariate Analysis, 71 (1999), no. 1, 67–75. http://dx.doi.org/10.1006/jmva.1999.1824 [15] K. Zografos and S. Nadarajah, Expressions for Rényi and Shannon entropies for multivariate distributions, Statistics & Probability Letters, 71 (2005), no. 1, 71–84. http://dx.doi.org/10.1016/j.spl.2004.10.023 Received: February 23, 2016; Published: April 1, 2016