A Bivariate Distribution whose Marginal Laws are Gamma and Macdonald

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1 International Journal of Mathematical Analysis Vol. 1, 216, no. 1, HIKARI Ltd, A Bivariate Distribution whose Marginal Laws are Gamma Macdonald Daya K. Nagar, dwin Zarrazola Luz stela Sánchez Instituto de Matematicas Universidad de Antioquia Calle 67, No. 5318, Medellin, Colombia Copyright c 216 Daya K. Nagar, dwin Zarrazola Luz stela Sánchez. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, reproduction in any medium, provided the original work is properly cited. Abstract Gamma Maconald distributions are associated with gamma extended gamma functions, respectively. In this article, we define a bivariate distribution whose marginal distributions are gamma Macdonald. We study several properties of this distribution. Mathematics Subject Classification: 3399, 65 Keywords: Confluent hypergeometric function; entropy; extended gamma function; gamma distribution; Laguerre polynomial 1 Introduction The gamma function was first introduced by Leonard uler in 1729 as the limit of a discrete expression later as an absolutely convergent improper integral, Γa) = t a 1 exp t) dt, Rea) >. 1) The gamma function has many beautiful properties has been used in almost all the branches of science engineering. Replacing t by z/σ, σ >,

2 456 Daya K. Nagar, dwin Zarrazola Luz stela Sánchez in 1), a more general definition of gamma function can be given as Γa) = σ a z a 1 exp z ) dz, Rea) >. 2) σ In statistical distribution theory, gamma function has been used extensively. Using integr 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), a >, σ >, v >. 3) σ a Γa) We will write V Ga, σ) if the density of V is given by 3). Here, a σ determine the shape scale of the distribution. In 1994, Chaudhry Zubair [3] defined the extended gamma function, Γa; σ), as Γa; σ) = t a 1 exp t σ ) dt, t where σ > a is any complex number. For Rea) > by taking σ =, it is clear that the above extension of the gamma function reduces to the classical gamma function, Γa, ) = Γa). The generalized gamma function extended) has been proved very useful in various problems in engineering physics, see for example, Chaudhry Zubair [2 6]. Using the integr of the extended gamma function, an extended gamma distribution can be defined by the p.d.f. v a 1 exp v σ/v), v >. Γa; σ) The distribution given by the above density will be designated as Ga, σ). By using the definition of the extended gamma function, Chaudhry Zubair [4] have introduced a one parameter Macdonald distribution. By making a slight change in the density proposed by Chaudhry Zubair [4], a three parameter Macdonald distribution Nagar, Roldán-Correa Gupta [7, 8]) is defined by the p.d.f. f M y; α, β, σ) = σ β y β 1 Γα; σ 1 y), y >, σ >, β >, α + β >. Γβ)Γα + β) We will denote it by Y Mα, β, σ). If σ = 1 in the density above, then we will simply write Y Mα, β). By replacing Γα; σ 1 y) by its integral representation, the three parameter Macdonald density can also be written as f M y; α, β, σ) = σ β y β 1 Γβ)Γα + β) exp x y ) x α 1 dx, y >, 4) σx

3 A bivariate distribution whose marginal laws are gamma Macdonald 457 where σ >, β > α + β >. Now, consider two rom variables X Y such that the conditional distribution of Y given X is gamma with the shape parameter β the scale parameter σx the marginal distribution of X is a stard gamma with the shape parameter α + β. That is fy x) = yβ 1 exp y/σx) Γβ)σx) β, y > gx) = xα+β 1 exp x), x >. Γα + β) Then 4) can be written as f M y; α, β, σ) = fy x)gx) dx. Thus, the product fy x)gx) can be used to create a bivariate density with Macdonald stard gamma as marginal densities of Y X, respectively. We, therefore, define the bivariate density of X Y as fx, y; α, β, σ) = xα 1 y β 1 exp x y/σx), x >, y >, 5) σ β Γβ)Γα + β) where β >, α + β > σ >. 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 the conditional distributions of Y are Macdonald gamma, the marginal distribution of X is gamma, 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 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 cidate for many real life problems. In this article, we study distributions defined by the density 5), derive properties such as marginal conditional distributions, moments, entropies, information matrix, distributions of sum quotient.

4 458 Daya K. Nagar, dwin Zarrazola Luz stela Sánchez 2 Properties Let us now briefly discuss the shape of 5). ln fx, y; α, β, σ) with respect to x y are The first order derivatives of f x x, y) = f y x, y) = ln fx, y; α, β, σ) x ln fx, y; α, β, σ) y = α 1 x = β 1 y + y σx 2 1 6) 1 σx 7) respectively. Setting 6) 7) to zero, we note that a, b), a = α + β 2, b = σβ 1)α + β 2) is a stationary point of 5). Computing second order derivatives of ln fx, y; α, β, σ), from 6) 7), we get f xx x, y) = 2 ln fx, y; α, β, σ) = α 1 2y x 2 x 2 σx, 8) 3 f xy x, y) = 2 ln fx, y; α, β, σ) x y = 1 σx 2, 9) f yy x, y) = 2 ln fx, y; α, β, σ) y 2 = β 1 y 2. 1) Further, from 8), 9) 1), we get f xx a, b) = α + 2β 3 α + β 2), f 1 yya, b) = 11) 2 σ 2 β 1)α + β 2) 2 f xx a, b)f yy a, b) [f xy a, b)] 2 = 1 σ 2 β 1)α + β 2) 3. 12) Now, observe that If β > 1 α+β > 2, then f xx a, b)f yy a, b) [f xy a, b)] 2 >, f xx a, b) < f yy a, b) < therefore a, b) is a maximum point. If β < 1 α+β < 2, then f xx a, b)f yy a, b) [f xy a, b)] 2 >, f xx a, b) > f yy a, b) > therefore a, b) is a minimum point. If β < 1 α + β > 2, then f xx a, b)f yy a, b) [f xy a, b)] 2 <, therefore a, b) is a saddle point.

5 A bivariate distribution whose marginal laws are gamma Macdonald 459 A distribution is said to be positively likelihood ratio dependent PLRD) if the density fx, y) satisfies fx 1, y 1 )fx 2, y 2 ) fx 1, y 2 )fx 2, y 1 ) 13) for all x 1 > x 2 y 1 > y 2. In the present case 13) is equivalent to y 1 x 2 + x 1 y 2 x 1 y 1 + x 2 y 2 which clearly holds. Olkin Liu [11] have listed a number of properties of PLRD distributions. By definition, the product moments of X Y associated with 5) are given by X r Y s ) = = σs Γβ + s) Γβ)Γα + β) x α+r 1 y β+s 1 exp x y/σx) dy dx σ β Γβ)Γα + β) x α+β+r+s 1 exp x) dx = σs Γβ + s)γα + β + r + s), 14) Γβ)Γα + β) where both the lines have been derived by using the definition of gamma function. For r = s, the above expression reduces to X s Y s ) = σs Γβ + s), 15) Γβ) which shows that Y/σX has a stard gamma distribution with shape parameter β. Substituting appropriately in 14), means variances of X Y the covariance between X Y are computed as X) = α + β, VarX) = α + β, Y ) = σβα + β), VarY ) = σ 2 βα + β)α + 2β + 1), CovX, Y ) = σβα + β). The correlation coefficient between X Y is given by β ρ X,Y = α + 2β + 1. The variance-covariance matrix Σ of the rom vector X, Y ) whose bivariate density is defined by 5) is given by [ ] 1 σβ Σ = α + β) σβ σ 2 βα + 2β + 1)

6 46 Daya K. Nagar, dwin Zarrazola Luz stela Sánchez Further, the inverse of the covariance matrix is given by Σ 1 = 1 σα + β)α + β + 1) [ ] σα + 2β + 1) βσ The well-known Mahalanobis distance is given by D 2 1 = [σα + 2β + 1)X 2 2XY + Y 2 σα + β)α + β + 1) βσ ] 2σα + β)α + β + 1)X + σα + β) 2 α + β + 1) with D 2 ) = 2 D 4 2 ) = βα + β)α + β + 1) [ ] βα + β)α + β + 4) + 3β + 1)α + β + 2)α + β + 3) From the construction of the bivariate density 5), it is cleat that Y Mα, β, σ), X Gα + β), Y x Gβ, σx) X y Gα, y/σ). Making the transformation S = X + Y R = Y/X + Y ) with the Jacobian Gy, x r, s) = s in 5), the joint density of R S is given by f R,S r, s; α, β, σ) = 1 r)α 1 r β 1 s α+β 1 exp [ s + rs r/σ1 r)], σ β Γβ)Γα + β) where < r < 1 s >. Now, integrating s by using gamma integral, the marginal density of R is derived as f R r; α, β, σ) = 1 r) β 1 r β 1 exp [ r/σ1 r)], < r < 1. σ β Γβ) From the above density it can easily be shown that R/σ1 R) = Y/σX has a stard gamma distribution with shape parameter β. By integrating r, the marginal density of S is derived as f S s; α, β, σ) = sα+β 1 exp s) σ β Γβ)Γα + β) 1 1 r) α 1 r β 1 exp [ ] r rs dr. σ1 r) Now, writing [ ] r exp = 1 r) σ1 r) ) 1 r m L m, σ m=

7 A bivariate distribution whose marginal laws are gamma Macdonald 461 where L m x) is the Laguerre polynomial of degree m, integrating r by using the integral representation of confluent hypergeometric function, we get the marginal density of S as f S s; α, β, σ) = sα+β 1 exp s) ) 1 1 L σ β m 1 r) α+1 1 r β+m 1 exp rs) dr Γβ)Γα + β) σ m= ) = Γα + 1)sα+β 1 exp s) σ β Γβ)Γα + β) m= L m 1 σ 1 F 1 β + m; α + β + m + 1; s), s >, where α + 1 >, β > σ >. 3 Central Moments Γβ + m) Γα + β + m + 1) By definition, the i, j)-th central joint moment of X, Y ) is given by µ ij = [X µ X ) i Y µ Y ) j ]. For different values of i j, expressions for µ ij are given by µ 3 = 2α + β), µ 21 = 2σβα + β), µ 12 = 2σ 2 βα + β)α + 2β + 1), µ 3 = 2σ 3 βα + β)[α + 2β + 1) 2 + β + 1)α + β + 1)], µ 4 = 3α + β)α + β + 2), µ 31 = 3σβα + β)α + β + 2), µ 22 = σ 2 βα + β)[3β + α + β + 1)α + 4β + 6)], µ 13 = 3σ 3 βα + β)[2β + 1)α + β + 1)α + 2β + 2) βα + β)α + 2β + 1)], µ 4 = 3σ 4 βα + β)[2β + 1)α + β + 1)α + 2β + 2)α + 2β + 3) βα + β)α + 2β + 1) 2 ], µ 5 = 4α + β)5α + 5β + 6), µ 41 = 4σβα + β)5α + 5β + 6), µ 32 = 4σ 2 βα + β)[4α α + β + 2)2α + 7β)]. 4 ntropies In this section, exact forms of Rényi Shannon entropies are determined for the bivariate distribution defined in Section 1.

8 462 Daya K. Nagar, dwin Zarrazola Luz stela 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 H SH f) the well-known Shannon entropy introduced in Shannon [13]. It is define by H SH f) = fx) ln fx) dµ. 16) X One of the main extensions of the Shannon entropy was defined by Rényi [12]. This generalized entropy measure is given by where H R η, f) = ln Gη) 1 η Gη) = for η > η 1), 17) X f η dµ. The additional parameter η is used to describe complex behavior in probability models 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 Zografos [1], Zografos Nadarajah [15] Zografos [14]. Now, we derive the Rényi 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 the Shannon entropies are given by H R η, f) = 1 [ln Γ[ηβ 1) + 1] + ln Γ[ηα + β 2) + 2] η 1) ln σ 1 η [ηα + 2β 3) + 3] ln η η ln Γβ) η ln Γα + β)] 18) H SH f) = [β 1)ψβ) + α + β 2)ψα + β) ln σ α + 2β) ln Γβ) ln Γα + β)]. 19) Proof. For η > η 1, using the p.d.f. of X, Y ) given by 5), we have Gη) = = [fx, y; α, β, σ)] η dy dx 1 [σ β Γβ)Γα + β)] η Γ[ηβ 1) + 1] = σ η2β 1)+1 η ηβ 1)+1 [Γβ)Γα + β)] η Γ[ηβ 1) + 1]Γ[ηα + β 2) + 2] = σ η 1 η ηα+2β 3)+3 [Γβ)Γα + β)], η x ηα 1) y ηβ 1) exp ηx ηy σx ) dy dx x ηα+β 2)+1 exp ηx) dx

9 A bivariate distribution whose marginal laws are gamma Macdonald 463 where, to evaluate above integrals, we have used the definition of gamma function. Now, taking logarithm of Gη) using 17), we get 18). The Shannon entropy 19) is obtained from 18) by taking η 1 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 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 ln Lα,β,σ) ) α 2 ) ) 2 ln Lα,β,σ) β α 2 ln Lα,β,σ) σ α ) 2 ln Lα,β,σ) β α 2 ln Lα,β,σ) 2 ln Lα,β,σ) σ β ) β 2 ) 2 ln Lα,β,σ) σ α ) ) 2 ln Lα,β,σ) β σ ) 2 ln Lα,β,σ) σ 2, where Lα, β, σ) = ln fx, y; α, β, σ). Lα, β, σ) is obtained as From 5), the natural logarithm of ln Lα, β, σ) = β ln σ ln Γβ) ln Γα + β) + α 1) ln x + β 1) ln y x y σx, where x > y >. The second order partial derivatives of ln Lα, β, σ) are given by = ψ α 2 1 α + β), β 2 σ 2 α β = ψ 1 β) ψ 1 α + β), = β σ 2 2y σ 3 x, = ψ 1 α + β), =, α σ = 1 β σ σ,

10 464 Daya K. Nagar, dwin Zarrazola Luz stela Sánchez where ψ 1 ) is the trigamma function. Now, noting that the expected value of a constant is the constant itself Y/σX follows a stard gamma distribution with shape parameter β, we have 6 stimation ) = ψ α 2 1 α + β), ) = ψ β 2 1 β) ψ 1 α + β), ) = β σ 2 σ, 2 ) = ψ 1 α + β), α β ) =, α σ ) = 1 β σ σ. 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 x 1, y 1 ),..., x n, y n ) is a rom sample from 5). The loglikelihood can be expressed as: ln Lα, β, σ) = nβ ln σ n ln Γβ) n ln Γα + β) + α 1) + β 1) ln y i x i + y ) i. σx i ln x i The first-order derivatives of this with respect to the three parameters are: ln Lα, β, σ) α = nψα + β) + ln x i,

11 A bivariate distribution whose marginal laws are gamma Macdonald 465 ln Lα, β, σ) β = n ln σ nψβ) nψα + β) + ln Lα, β, σ) σ = nβ σ + 1 σ 2 y i x i. ln y i, The maximum likelihood estimators of α, β, σ), say ˆα, ˆβ, ˆσ), are the simultaneous solutions of the above three equations. It is straightforward to see that ˆβ can be calculated by solving numerically the equation [ ψ ˆβ) ln ˆβ = 1 ) )] yi y i ln ln. n Further, for ˆβ so obtained, ˆσ ˆα can be computed by solving numerically the equations ˆβˆσ = 1 y i n ψˆα + ˆβ) = 1 n x i x i ln x i respectively. Using the expansion of the digamma function, namely, ψz) = ln z z 1 48 z z z 199 ) z + 5 an approximation for ˆβ can be given as ˆβ = 1 [ ] 1 q 2 n q) 1, 1/n where q = n q i/n, q = n q i) 1/n q i = y i /x i, i = 1,..., n. Using this estimate of β, the estimates of σ α are given by [ ˆσ = 2 q q n q) 1/n 1 ˆα = x 1 ] 1 [1 n q)1/n, 2 q respectively, where x = n x i) 1/n. Acknowledgements. The research work of DKN LS was supported by the Sistema Universitario de Investigación, Universidad de Antioquia under the project no. IN1231C. ] x i

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13 A bivariate distribution whose marginal laws are gamma Macdonald 467 [11] Ingram Olkin Ruixue Liu, A bivariate beta distribution, Statistics & Probability Letters, 62 23), no. 4, [12] A. Rényi, On measures of entropy information, Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics Probability, Vol. I, Univ. California Press, Berkeley, Calif., 1961), [13] C.. Shannon, A mathematical theory of communication, Bell System Technical Journal, ), , [14] K. Zografos, On maximum entropy characterization of Pearson s type II VII multivariate distributions, Journal of Multivariate Analysis, ), no. 1, [15] K. Zografos S. Nadarajah, xpressions for Rényi Shannon entropies for multivariate distributions, Statistics & Probability Letters, 71 25), no. 1, Received: February 23, 216; Published: April 1, 216