BIVARIATE NONCENTRAL DISTRIBUTIONS: AN APPROACH VIA THE COMPOUNDING METHOD

Transcription

1 South African Statist J 06 50, BIVARIATE NONCENTRAL DISTRIBUTIONS: AN APPROACH VIA THE COMPOUNDING METHOD Johan Ferreira Departent of Statistics, Faculty of Natural and Agricultural Sciences, University of Pretoria, Pretoria, South Africa e-ail: ohanferreira@upacza Andriette Bekker Departent of Statistics, Faculty of Natural and Agricultural Sciences, University of Pretoria, Pretoria, South Africa Mohaad Arashi Departent of Statistics, School of Matheatical Sciences, University of Shahrood, Shahrood, Iran Departent of Statistics, Faculty of Natural and Agricultural Sciences, University of Pretoria, Pretoria, South Africa Key words: Bivariate chi-square distribution, bivariate F distribution, coposites, copounding, noncentral, Poisson distribution Abstract: This paper enriches the existing literature of bivariate noncentral distributions by proposing bivariate noncentral generalised chi-square and F distributions via the eployent of the copounding ethod with Poisson probabilities This ethod has been used to a liited extent to obtain univariate noncentral distributions; this study extends soe results in literature to the corresponding bivariate setting The process which is followed to obtain such bivariate noncentral distributions is systeatically described and otivated Soe distributions of coposites univariate functions of the dependent coponents of the bivariate distributions are derived, in particular the product, ratio, and proportion Furtherore, an exaple of possible application is given and discussed to illustrate the versatility of the proposed odels Introduction Bivariate distributions can be constructed in a variety of ways fro univariate settings see Balakrishnan and Lai, 009 Patnaik 949 showed that the noncentral chi-square distribution with n degrees of freedo and noncentrality paraeter θ can be represented as a weighted su of univariate chi-square probabilities with weights equal to the probabilities of a Poisson distribution Corresponding author AMS: 6F5, 6H, 6H0, 60E05

2 04 FERREIRA, BEKKER, & ARASHI with expected value θ In the text by Johnson, Kotz and Balakrishnan 995, p 433, the noncentral chi-square distribution is presented as follows: Let X n i U i, where U,U,,U n are independent rando variables and U i is norally distributed with ean µ i and unit variance Then the probability density function pdf of a noncentral chi-square distributed rando variable X with n degrees of freedo, and noncentrality paraeter θ n i µ i is given by f X x k0 x n +k e x n +k Γ n + k e θ θ k, x > 0 k! where n > 0 and θ > 0 This pdf can be viewed as a copound pdf: f X x k0 f X x kg K k such that f X x k x n +k e x n +k Γ n +k, x > 0, and g Kk e θ θ k k!, k 0,,,3, The genesis for this paper originated fro the papers by Yunus and Khan 0 and Van den Berg, Roux and Bekker 03, with Khan, Pratikno, Ibrahi and Yunus 05 aking recent valuable contributions In this paper the authors use the above univariate description as a departure point for constructing an analogy to the bivariate setting Yunus and Khan 0 entioned that the copounding ethod to derive distributions by ixing ore than one distribution is well known in the literature; in this paper, we propose the following: Description An unconditional bivariate noncentral pdf can be obtained fro a conditional bivariate central pdf in the following anner: θv Here, g Kv k v e θv kv f X,X x,x k 0 k 0 f X,X x,x k,k g K k g K k k v!,v, is the Poisson weights where θ v are the noncentrality paraeters, and f x,x k,k being the pdf of soe suitable conditional bivariate central distribution for X and X By using this ethod a new representation of a bivariate noncentral generalised chi-square distribution is proposed here, shown to be equivalent to an existing distribution by showing that their respective oent generating functions are equal The construction of this distribution is systeatically described and outlined Subsequently, a bivariate noncentral generalised F distribution is derived Furtherore, soe univariate distributions of coposites a univariate function of the coponents of the bivariate distribution are derived for both the bivariate noncentral generalised chisquare distribution, and its F counterpart Al-Ruzaiza and El-Gohary 008 discuss the relevance and application of such univariate distributions, considering their natural occurrence in, aongst others, genetics, nuclear physics, and eteorology The coposites under consideration in this paper are, the product of the coponents W X X, the ratio of the coponents W X X tered, the ratio of type II, and the proportion of the coponents W 3 X X +X tered, the ratio of type I In Section, this alternative representation of an existing bivariate noncentral generalised chisquare distribution is systeatically derived and otivated Section 3 includes the developent

3 BIVARIATE NONCENTRAL DISTRIBUTIONS 05 in a siilar way of a bivariate noncentral generalised F distribution, along with an expression for its products oents Soe univariate distributions of the coposites are derived for both chisquare and F distributions considered in this paper in Section 4 An application of derived results is proposed in Section 5, and conclusions are discussed in Section 6 An alternative ethod In this section, an alternative expression for an existing bivariate noncentral generalised chi-square distribution is proposed This distribution is systeatically constructed by using the following approach: a eploy the bivariate central generalised chi-square distribution as defined by Van Den Berg 00, and ipose conditions on the degrees of freedo; b reove the iposed condition by using Poisson weights; and c the resulting pdf is that of the corresponding bivariate noncentral generalised chi-square distribution Furtherore, we show that the oent generating function gf of the new proposed distribution is identical to that of the one derived in Van Den Berg 00 The section concludes with a brief discussion regarding the obtained results Probability density function Suppose X and X are rando variables, each having a central chi-square distribution with n and n degrees of freedo respectively The oint pdf is constructed such that the arginal distributions are correlated Van Den Berg 00 defined a oint gf of X and X of the for M X,X t,t t n t n where n r and n r, with the pdf of this distribution given as n f X,X x,x x e x x n L n e x x L n 4ξ r t t t t r 0 x!ξ n +n Γ n + Γ n +, x,x > 0 3 where n,n,r > 0, n r and n r, ξ, and L n is the Laguerre polynoial as defined in Kotz, Balakrishnan, Read, Vidakovic and Johnson 006, p 9 This oint distribution of X and X see 3 is referred to as the bivariate generalised chi-square distribution ξ is the correlation coponent, r is an additional for paraeter, and Γ is the gaa function Along with the bivariate central generalised chi-square gf defined by Van Den Berg 00, an gf for a bivariate noncentral generalised chi-square distribution was also defined, given by M X,X t,t t n t n 4ξ r t t t t e θ t t e θ t t, t,t > 0 4

4 06 FERREIRA, BEKKER, & ARASHI This distribution also contains the additional paraeter r, and the corresponding pdf is given by n f X,X x,x x e x x +k! +k!!k!k! r θ k θ k 0 k 0 k 0 Γ n + + k Γ n + + k ξ n n L n +n +k +k +k x L +k x, x,x >0 n e x where n,n,r > 0, and n r and n r The paraeter ξ, where ξ, is a coponent of the product-oent correlation between X and X The paraeters θ,θ > 0 are the noncentrality paraeters respectively, and the arginal distribution of X and X are univariate noncentral chi-square distributions with paraeters n,θ and n,θ see Van Den Berg, 00 Consider Description in the introduction of this paper To this end, a conditional bivariate central chi-square pdf needs to be defined and is considered next Definition Let X,X K k,k k have a conditional bivariate generalised chi-square distribution see 3 with pdf given by n +k f X,X x,x k,k x e x x n +k e x 0 r!ξ n +n +k +k n L +k n +k x L x Γ n + + k Γ n, x,x > k where n,n,r > 0 and n r and n r As previously, ξ where ξ is a paraeter which is a coponent of the product-oent correlation between X and X The conditional values k and k have doain such that k v 0, v, By substituting 6 as the conditional bivariate distribution in, an alternative representation to the bivariate noncentral generalised chi-square distribution see 5 is proposed Definition The oint pdf of X and X, that is an alternative representation of the bivariate noncentral generalised chi-square pdf 5, is proposed by substituting 6 in, with the following result: 5 f X,X x,x k 0 k 0 0 k 0 k 0 r e θ f X,X x,x k,k g K k g K k n +k x e x x!ξ n +n +k +k k θ e θ k! n +k e x L n +k n +k x L x Γ n + + k Γ n + + k k θ, x,x > 0 7 k!

5 BIVARIATE NONCENTRAL DISTRIBUTIONS 07 where n,n,r > 0, n r and n r, ξ, and noncentrality paraeters θ,θ > 0 Evidently, the Poisson weights, naely g Kv k v e θ v/ θ v / k v /k v!,v,, isolate the noncentrality paraeters as suggested by in a atheatically convenient way In Figure, the contour plots for the distribution in 7 are given to illustrate its for for arbitrary paraeter values: n 0, n, and r In Figure, θ 3 reains fixed together with ξ 05, whilst θ varies Figure : Fro left to right, 7 for θ 3, and θ 5, 8, and For this change in θ, it is seen that as θ increases, the pdf oves away fro the axis of variable X which is to be expected, as θ represents the noncentrality of variable X Note that the sae effect would be observed for changes in θ, but oving away fro the axis of variable X, because of the syetric nature of the noncentrality coponents Poisson weights Fro 7, consider the oent generating function: M X,X t,t E e t X +t X 0 k 0 k e t x n +k x r e t n x +k x n +k L!ξ Γ n + k + Γ n + k + n +k L e θ x dx x dx k θ e θ k! n +n +k +k k θ k! By applying Prudnikov, Brychkov and Marichev 986b, eq 933, p 46 to both integrals above, one obtains: M X,X t,t 0 k 0 k 0 ξ r! t n +k t n +k 4t t t t e θ k θ e θ k θ k! k!

6 08 FERREIRA, BEKKER, & ARASHI Now, by reordering the expression accordingly and rearranging constant ters, the following is obtained: M X,X t,t t n t n 4ξ t t t t k k θ θ t k! e θ t k 0 k! e θ k 0 By using the binoial expansion and the series expansion for the exponential function, the oint gf is given by r M X,X t,t t n t n 4 e θ t t e θ t t, t,t > 0 4ξ r t t t t Therefore, this distribution with pdf defined in 7 is an alternative representation of the bivariate noncentral generalised chi-square distribution given in 5, ie 5 7 Reark Note that when θ θ 0 in 4 the expression reduces to that of the oent generating function of a bivariate central chi-square distribution see This otivates the naing convention of the bivariate noncentral chi-square distribution in 7 3 A bivariate noncentral generalised F distribution Since we have now seen that the proposal of the copounding idea works for the bivariate noncentral generalised chi-square distribution, in this section we will apply the ethod given in Description and define a new bivariate noncentral generalised F distribution This new distribution is derived by deriving a bivariate central generalised F distribution via the transforation approach fro its chi-square counterpart see 3 Next in a siilar anner as in Section, the systeatic construction of its noncentral counterpart is then proposed by defining a conditional bivariate central generalised F distribution, and the condition subsequently reoved with Poisson weights to obtain the noncentral distribution 3 Probability density function A bivariate central generalised F distribution is derived in this section via the transforation approach, by using the bivariate central generalised chi-square distribution in 3 Theore Let X and X be ointly distributed with pdf 3 with n and n degrees of freedo respectively, and let Z χ be an independent chi-square distributed rando variable with

7 BIVARIATE NONCENTRAL DISTRIBUTIONS 09 degrees of freedo Let Y,Y X /n Z /, X / n Z / The oint pdf of Y and Y is given by f Y,Y y,y 0 l 0 l 0 ξ r l +l l! Γ n +n + + l + l Γ n + l Γ n + l Γ n n +l y +l y l n + n y + n y n +l n n +l n +n + +l +l, y,y > 0 8 where n,n,r, > 0, and ξ Proof The proof follows directly following a transforation Definition 3 Let Y,Y K k,k k have a conditional bivariate generalised F distribution see 8 with pdf given by f Y,Y y,y k,k ξ r 0 l 0 l 0 n l +l l! l n +l +k n n +l +k n +l y +k + n y + n y Γ n +n + + l + l + k + k Γ n + l + k Γ n + l + k Γ n +l y +k n +n + +l +l +k +k, y,y > 0 9 where n,n,r, > 0 As previously, ξ where ξ is a paraeter which is a coponent of the product-oent correlation between X and X The conditional values k and k have doain such that k,k 0 Siilar to Section, a bivariate noncentral generalised F distribution is obtained by applying Description for 8 together with the respective Poisson weights, and is defined next Definition 4 The oint pdf of Y and Y, which represents the bivariate noncentral generalised F distribution is proposed by substituting 9 in, with the following result: f Y,Y y,y k 0 k 0 f Y,Y y,y k,k g K k g K k 0 k 0 k 0 l 0 l 0 + n e θ y + n y k θ e θ k! n +l Cy +k y n +l +k n +n + +l +l +k +k k θ, y,y > 0 0 k!

8 0 FERREIRA, BEKKER, & ARASHI where C l +l l l ξ r Γ n +n + + l + l + k + k! Γ n + l + k Γ n + l + k Γ n n +l +k n n +l +k and n,n,r, > 0, and θ,θ > 0 Again, due to the construction of the noncentrality, the Poisson probability factors, naely g Kv k v e θ v/ θ v / k v /k v!,v,, isolates the noncentrality paraeters in a atheatically convenient way In Figure, the contour plots for the distribution in 0 are given to illustrate its for for arbitrary paraeter values: n 0, n, and r In Figure, θ 3 reains fixed together with ξ 05, whilst θ varies For the change in θ, it is seen that as θ increases, the pdf becoes ore Figure : Fro left to right, 0 for θ 3, and θ 5, 8, and attracted to the axis of the corresponding variable X Siilar to the shape analysis of the analogous bivariate chi-square distribution in Section, this is to be expected Reark In this section a bivariate noncentral generalised F distribution is proposed see 0 by using the copounding ethod see after defining a conditional bivariate generalised F distribution in 9 Note that the sae bivariate distribution 0 would have been obtained as a oint distribution for Y and Y with a transforation approach had the oint pdf of X and X been the bivariate noncentral generalised chi-square distribution as given in 7, together with Z χ independent and Y X / n Z / and Y X / n Z / 3 Product oents In the following theore, an expression for the product oents is derived Theore If Y and Y are ointly distributed according to 0, then the product oent, ie E Y q Y s, is given by

9 BIVARIATE NONCENTRAL DISTRIBUTIONS E Y q Y s 0 k 0 k 0 l 0 l 0 l +l l l ξ r! Γ n +n + + k + k + l + l Γ n + k + l Γ n + k + l Γ n n q s B s + n + k + l, n + + k + l s θ B q + n + k + l, e q s k θ e θ k! θ k k! where n,n,,r > 0, ξ, and θ,θ > 0 B, is the beta function with values of arguent such that B, is well-defined Proof Fro 0 and C is the value defined in : E Y q Y s 0 0 k 0 k 0 y q+ n +l +k 0 k 0 k 0 l 0 l 0 B s+ n l 0 0 l 0 e θ C k θ k! y s+ n +l +k e θ θ k k! n + n n y + y e θ k θ e θ k θ C k! k! n +l +k ; n + +l +k s 0 y q+ n +l +k n +n + +l +l +k +k n n +n + +l +l +k +k n +n + +l +l +k +k + n y n dy dy n + +l +k s dy which follows by applying Prudnikov, Brychkov and Marichev 986a, eq 44, p 98: setting α s + n + l + k, ρ n +n + + l + l + k + k, and z n + n n y The reaining integral is solved by applying the sae result: setting then α q + n + l + k, ρ n + + l + k s, and z n ; and then, after soe siplification, the proof is coplete 4 Distributions of coposites The following two subsections contain the derivations of the pdfs for the univariate distributions of coposites; specifically, the product, the ratio of type II, and the ratio of type I These coposite pdfs are derived for both their bivariate noncentral generalised chi-square- and F counterparts Siilar to previous sections, the derivation of the coposite pdfs will be described systeatically: first obtaining the result for the conditional central case, and subsequently unconditioning the conditional pdf to obtain the corresponding noncentral distribution counterpart

10 FERREIRA, BEKKER, & ARASHI 4 Chi-square distribution setting see 7 Here the results as derived by Van Den Berg 00 are presented as Result A, B, and C respectively A siilar approach as in Section is followed where a conditional case is defined, and subsequently unconditioned to obtain its noncentral counterparts Result A If X and X are ointly distributed according to 3, the pdf of W X X is given by f W w f W w 0 l 0 l 0 0 l 0 l 0 K τ w w n + n +l +l n +n +l +l, w > 0, n +l w +w n +n +l +l ξ r l +l l!γ n + l Γ n + l where n,n,r > 0, ξ, τ n n + l l, and K τ the odified Bessel function of the second kind see Gradshteyn and Ryzhik, 007, eq 8433, p 97 Result B If X and X are ointly distributed according to 3, the pdf of W X X is given by ξ r l +l l l!b n +l, n +l l, w > 0, where n,n,r > 0, ξ Result C If X and X are ointly distributed according to 3, the pdf of W 3 X X +X is given by n +l f W3 w 3 w 3 w 3 n +l ξ r l +l l l 0 l 0 l 0!B n +l, n, 0 < w 3 <, +l where n,n,r > 0, ξ 4 The probability density function of the product In this section, the pdf of the product of the coponents of the bivariate distribution proposed in 6 is given Subsequently the noncentral case is proposed by using the copounding ethod Let W X X Theore 3 If X and X are ointly distributed according to 6, the pdf of the conditional distribution of W K k,k k is given by f W w k,k K τ n + n +l +l +k +k r w w ξ! 0 l 0 l 0 l +l l l Γ n + l + k Γ n + l + k n +n +l +l +k +k, w > 0, where n,n,r > 0, ξ, τ n n +l l +k k, and K τ the odified Bessel function of the second kind see Gradshteyn and Ryzhik, 007, eq 8433, p 97 The conditional values k and k have doain such that k v 0, v,

11 BIVARIATE NONCENTRAL DISTRIBUTIONS 3 Proof See Result A, Section 4 Corollary Upon taking the pdf in one can now obtain the unconditional noncentral distribution of W X X by substituting the Poisson weights and the corresponding suation operators siilar to : f W w f W w k,k g K k g K k 0 k 0 k 0 l 0 l 0 K τ w w l +l l l Γ n + l + k Γ n + l + k e θ k θ e θ k! k θ, w > 0, k! n + n +l +l +k +k r ξ! n +n +l +l +k +k where n,n,r > 0, ξ, τ n n + l l + k k, θ,θ > 0, and g Kv k v e θ v/ θ v / k v /k v!, v, 4 The probability density function of the ratio of type II In this section, the pdf of the ratio of type II of the coponents of the bivariate distribution proposed in 6 is given Subsequently the noncentral case is proposed by using the copounding ethod Let W X X Theore 4 If X and X are ointly distributed according to 6, the pdf of the conditional distribution of W K k,k k is given by f W w k,k 0 l 0 l 0 n +l w +k r ξ + w τ! l +l l l B n + l + k, n, w > 0, 3 + l + k where n,n,r > 0, ξ, and τ n k have doain such that k v 0, v, + n + l + l + k + k The conditional values k and Proof See Result B, Section 4 Corollary Upon taking the pdf in 3 one can now obtain the unconditional noncentral distribution of W X X by substituting the Poisson weights and the corresponding suation operators

12 4 FERREIRA, BEKKER, & ARASHI siilar to : f W w f W w k,k g K k g K k 0 k 0 k 0 l 0 l 0 n +l +k w + w τ l +l l l B n + l + k, n + l + k e θ where n,n,r > 0, ξ, τ n e θv/ θ v / k v /k v!, v, k θ e θ k! + n 43 The probability density function of the ratio of type I r ξ! k θ, w > 0, 4 k! + l + l + k + k, θ,θ > 0, and g Kv k v In this section, the pdf of the ratio of type I of the coponents of the bivariate distribution proposed in 6 is given Subsequently the noncentral case is proposed by using the copounding ethod Let W 3 X X +X Theore 5 If X and X are ointly distributed according to 6, the pdf of the conditional distribution of W 3 K k,k k is given by n r +l f W3 w 3 k,k w +k 3 w 3 n +l +k ξ! 0 l 0 l 0 l +l l l B n + l + k, n + l + k, 0 < w 3 <, 5 where n,n,r > 0, and ξ The conditional values k and k have doain such that k v 0, v, Proof See Result C, Section 4 Corollary 3 Upon taking the pdf in 5 one can now obtain the unconditional noncentral distribution of W 3 X X +X by substituting the Poisson weights and the corresponding suation operators siilar to : f W3 w 3 f W3 w 3 k,k g K k g K k 0 k 0 k 0 l 0 l 0 r e θ ξ! k θ e θ k! n +l w +k 3 w 3 n +l +k l +l l l B n + l + k, n + l + k k θ, 0 < w 3 <, k!

13 BIVARIATE NONCENTRAL DISTRIBUTIONS 5 where n,n,r > 0, ξ, θ,θ > 0, and g Kv k v e θ v/ θ v / k v /k v!, v, 4 F-distribution setting see 0 In this section, the univariate distribution of the coposites of the product, the ratio of type II and the ratio of type I is derived for the distribution in 0 The approach again is systeatic by first deriving the conditional counterparts using the distribution in 9, and subsequently unconditioning using the copounding ethod 4 The probability density function of the product In this section, the pdf of the product of the coponents of the bivariate distribution proposed in 9 is derived Subsequently, the noncentral case is proposed by using the copounding ethod as in Description Let W Y Y Theore 6 If Y and Y are ointly distributed according to 9, the pdf of the conditional distribution of W K k,k k is given by f W w k,k 0 l 0 l 0 Cw 4 + n F n + 4 +l +k, n n + 4 +l +k n n + 4 +l +k + 4 +l +k ; n +n ++ +l +l +k +k ;, w >0 4w n n where n,n, > 0, ξ, and F, ; ; is the Gauss hypergeoetric function see Mathai, 993, p 96 with 4w n n < The conditional values k and k have doain such that k v 0, v,, and C is the value as given in Proof The Jacobian of the transforation is given by y, and thus fro 9: f W,Y w,y k,k 0 l 0 l 0 n +l Cw +k n y + y + n w Now, since y > 0; the conditional pdf of W is given by f W w k,k 0 l 0 l 0 Cw n +l +k 0 n y + y + n w y n + +l +k n +n + +l +l +k +k y n + +l +k n +n + +l +l +k +k dy where this last integral is evaluated using Prudnikov et al 986a, eq 97, p 309: setting ρ n +n + + l + l + k + k, α n + + l + k, a n, b, and c n w, the proof is coplete 6

14 6 FERREIRA, BEKKER, & ARASHI Corollary 4 Upon taking the pdf in 6 one can now obtain the unconditional noncentral distribution of W Y Y by substituting the Poisson weights and the corresponding suation operators siilar to : f W w f W w k,k g K k g K k n C 0 k 0 k 0 l 0 l 0 n + 4 +l +k n n F + 4 +l +k, n + 4 +l +k ; n +n ++ e θ k θ e θ k θ, w > 0, k! k! n + 4 +l +k w 4 + +l +l +k +k ; 4w n n where n,n, > 0, ξ, θ,θ > 0, and g Kv k v e θv/ θ v / k v /k v! for v,, and 4w n n < 4 The probability density function of the ratio of type II Here the pdf of the ratio of type II of the coponents of the bivariate distribution proposed in 9 is derived Subsequently the noncentral case is proposed by using the copounding ethod Let W Y Y Theore 7 If Y and Y are ointly distributed according to 9, the pdf of the conditional distribution of W K k,k k is given by f W w k,k 0 l 0 l 0 Cw n +l +k w n + n n + n B + l + l + k + k, n +n +l +l +k +k, w > 0, 7 where n,n, > 0, and ξ The conditional values k and k have doain such that k v 0, v,, and C is the value as given in Proof The Jacobian of the transforation is given by y, and thus fro 9 the oint conditional pdf of W and Y is given by f W,Y w,y k,k n +l Cw +k 0 l 0 l 0 y + n w + n y n +n +l +l +k +k n +n + +l +l +k +k Now, since y > 0; the pdf of W is given by

15 BIVARIATE NONCENTRAL DISTRIBUTIONS 7 f W w k,k 0 l 0 l 0 0 n +n y n +l Cw +k +l +l +k +k + n w + n y n +n + +l +l +k +k dy The latter integral is siplified by applying Gradshteyn and Ryzhik 007, eq 3943, p 35: setting β n w +n, µ n +n +l +l +k +k, κ n +n + +l +l +k +k, and by substituting the result the proof is coplete Corollary 5 Upon taking the pdf in 7 one can now obtain the unconditional noncentral distribution of W Y Y by substituting the Poisson weights and the corresponding suation operator siilar to : f W w f W w k,k g K k g K k 0 k 0 k 0 l 0 l 0 Cw n +l +k n + n B + l + l + k + k, e θ k θ e θ k! n w + n n +n +l +l +k +k k θ, w > 0, 8 k! where n,n, > 0, ξ, θ,θ > 0, and g Kv k v e θ v θ v / k v /k v! for v, 43 The probability density function of the ratio of type I Here the pdf of the ratio of type I of the coponents of the bivariate distribution proposed in 9 is derived Subsequently the noncentral case is proposed by using the copounding ethod Let W 3 Y Y +Y Theore 8 If Y and Y are ointly distributed according to 9, the pdf of the conditional distribution of W 3 K k,k k is given by f W3 w 3 k,k 0 l 0 l 0 n +l Cw +k 3 w 3 n +l +k + n + n B + l + l + k + k, w n 3 w 3 + n n +n +l +l +k +k, 0 < w 3 <, 9 where n,n, > 0, and ξ C is the value as given in, and the conditional values k and k have doain such that k v 0, v,

16 8 FERREIRA, BEKKER, & ARASHI Proof Consider the following: if W 3 Y Y +Y, and W Y Y, then W W 3 W 3 Furtherore, the Jacobian of this transforation is given by dw dw 3, and it follows that w 3 f W3 w 3 f W w3 w 3 dw dw 3 By using this result and via substitution, the following is obtained: f W3 w 3 k,k 0 l 0 l 0 n w3 +l +k w n 3 w C 3 + n w 3 n + n B + l + l + k + k, which copletes the proof n +n +l +l +k +k w 3 Corollary 6 Upon taking the pdf in 9 one can now obtain the unconditional noncentral distribution of W 3 Y Y +Y by substitution the Poisson weights and the corresponding suation operators siilar to : f W3 w 3 f W3 w 3 k,k g K k g K k 0 k 0 k 0 l 0 l 0 n + n B + l + l + k + k, n +l Cw +k 3 w 3 n +l +k + e θ θ k k! e θ w n 3 w 3 + n n +n +l +l +k +k k θ, 0 < w 3 <, k! where n,n, > 0, ξ, θ,θ > 0, and g Kv k v e θ v/ θ v / k v /k v! for v, 5 Application In this section, an application of soe results fro the bivariate noncentral generalised chi-square distribution and the bivariate noncentral generalised F distribution is presented The data used is drought data fro the state of Nebraska, USA, obtained freely fro the website which consists of droughtand nondrought duration in onths for eight cliate divisions of Nebraska fro January 895 to Deceber 004 Specifically, the univariate distribution of the ratio of type II will be considered here see 8 Soe percentage points are also calculated for the univariate distribution of the ratio of type II of the bivariate noncentral generalised chi-square distribution see 3 Nebraska is divided into eight cliate divisions there is no 4 th division Considering the state as a whole, Nebraska is considered to have two aor cliate divisions: the eastern half of the state has a huid, continental cliate ie divisions 3, 6, and 9, and the western half of state which has

17 BIVARIATE NONCENTRAL DISTRIBUTIONS 9 a sei-arid cliate ie divisions,, and 7 Overall, it is known that the entire state encounters wide seasonal changes in teperature and precipitation throughout the year Figure 3 illustrates the outlay of Nebraska as divided by state Figure 3: Cliate divisions of Nebraska, USA Due to these known differences between the cliate divisions fro the eastern- and western half of the state, one would expect the nuber of dry onths between, say, division and division 9, to be independent fro each other This is ainly attributed to the fact that these divisions fall within different cliate regions, each with their own cliatic structure, and is under the assuption that Y nuber of dry onths for division, and Y nuber of dry onths for division 9, is ointly distributed according to the bivariate noncentral generalised F distribution It is our intention to show that the correlation coponent in the bivariate noncentral generalised F distribution s considered coposite see 8 will exhibit this independence in the estiation of the paraeters, by being negligibly sall The divisions that will be considered for this purpose are regions and 9 Furtherore, letting Y nuber of dry onths of division t, and Y nuber of nondry onths of division t, the distribution of W would give an indication of the degree of dryness experienced by the division in question: if W >, it iplies that Y > Y, or rather, that the division sees to statistically exhibit ore dry onths than otherwise This is investigated for divisions and 8 5 Model description and paraeter estiation The distributions of particular interest here are the following cases:

18 0 FERREIRA, BEKKER, & ARASHI the ratio of drought duration of division to drought duration of division 9 W drought duration of division / drought duration of division 9 Y Y ; the ratio of drought duration of division t to nondrought duration of division t W drought duration of division t / nondrought duration of division t Y Y t,8 By using the ethod of axiu likelihood for estiation of the paraeters, the paraeters of the distributions of W was calculated for the described cases and The log-likelihood functions were optiized by using the SAS/IML call nlpnra - this call finds the axiu value of the provided function via a Newton-Raphson algorith For the ratio of drought duration of division to drought duration of division 9, the data was fitted to the distribution given in 8 by using the above entioned procedure in SAS/IML Note that the paraeters n and n are assued known as it corresponds to the respective saple sizes, and was not estiated By considering these estiated values, it is observed that the correlation coponent, Table : Paraeter estiates of W, 8 for case Paraeter estiates Cliate division & 9 n 74 n 74 ˆ ˆr ˆξ ˆθ ˆθ AIC ˆξ, is extreely sall The expected independence entioned at the start of Section 5 sees to be confired by these results The coputed value of the Akaike Inforation Criterion AIC is also provided in Table In the second considered case drought versus nondrought duration for each considered division, the paraeter estiates are given in Table Note again that the paraeters n and n are assued known as it corresponds to the respective saple sizes, and was not estiated Siilar to before, it is observed that the estiated correlation coponent ˆξ is negligibly sall The independence for this case exhibits the independence of drought duration and nondrought duration, which is also expected within each cliate division Consider the generalised odel, ie 8, for division 8, the value of PW > PW < P Y Y <, the probability that the drought duration for division 8 will be greater than that of the nondrought duration This value is calculated, using the package Matheatica, as PW > PW < This sees to be a reasonable observation since division 8 lies ore to the east of the state - ie lies ore toward the huid / continental cliatic side of the state, therefore resulting in less drought-ridden onths than otherwise

19 BIVARIATE NONCENTRAL DISTRIBUTIONS Table : Paraeter estiates of W, 8 for case Paraeter estiates Cliate division 8 n n ˆ ˆr ˆξ ˆθ ˆθ AIC Percentage points of distribution Certain percentage points w α of the distribution of W X X are obtained nuerically by solving the equation w α 0 f W w dw α By considering 4 soe lower percentage points are calculated for arbitrary paraeters Siilar tabulations can be obtained for other values of the paraeters The calculated values are given in Table 3 This distribution is considered since it ay offer an alternative approach to the well-known stress-strength odel in the context of reliability, where the lifetie of a rando coponent with strength X is subected to a rando stress X The easure PX < X is thus of interest, and translates to PX /X < PW < thereby revealing the relevance of this specific distribution Table 3: Percentage points for W 4, for n 0, n, and r ξ θ θ α Conclusion This paper explored the use of the copounding ethod as a distributional building tool to obtain bivariate noncentral distributions The process of obtaining a bivariate noncentral generalised chi-square distribution fro a conditional existing bivariate central generalised distribution was systeatically described and otivated The newly obtained noncentral distribution was shown to be equivalent to the bivariate noncentral generalised chi-square distribution of Van Den Berg 00 by showing their respective gfs to be equal Furtherore, the corresponding bivariate noncentral generalised F distribution was derived in a siilar systeatic way In both these cases it is evident

20 FERREIRA, BEKKER, & ARASHI that the constructed for of the distribution isolates the noncentrality paraeters continuously in a atheatical convenient way; by retaining the in Poisson probability for Subsequently the product, ratio, and proportion of coponents in both chi-square and F distributions were derived An application to drought of Nebraska, USA, was given which illustrated the versatility of the newly proposed odels 7 Acknowledgeents The authors would like to hereby acknowledge the support of the StatDisT group This work is based upon research supported by the National Research foundation, Grant Re:CPRR No 9497 References AL-RUZAIZA, A S AND EL-GOHARY, A 008 The distributions of sus, products, and ratios of inverted bivariate beta distribution Applied Matheatical Sciences, 48, BALAKRISHNAN, N AND LAI, C 009 Continuous Bivariate Distributions nd edition Springer: New York GRADSHTEYN, I S AND RYZHIK, I M 007 Table of Integrals, Series and Products Acadeic Press: San Diego JOHNSON, N L, KOTZ, S, AND BALAKRISHNAN, N 995 Continuous Univariate Distributions, volue Wiley: New York KHAN, S, PRATIKNO, B, IBRAHIM, A I N, AND YUNUS, R M 05 The correlated bivariate noncentral F distribution and its application Counications in Statistics - Siulation and Coputation DOI:0080/ KOTZ, S, BALAKRISHNAN, N, READ, C B, VIDAKOVIC, B, AND JOHNSON, N L 006 Encyclopedia of Statistical Sciences, volue 7 nd edition Wiley: New York MATHAI, A M 993 A Handbook of Generalised Special Function for Statistical and Physical Sciences Clarendon Press: Oxford, UK PATNAIK, P 949 The noncentral chi-square and F distributions and their applications Bioetrika, 36, 0 3 PRUDNIKOV, A P, BRYCHKOV, Y A, AND MARICHEV, O I 986a Integrals and Series, Volue : Eleentary Functions, volue Gordon and Breach Publishers: New York PRUDNIKOV, A P, BRYCHKOV, Y A, AND MARICHEV, O I 986b Integrals and Series, Volue : Special Functions, volue Gordon and Breach Publishers: New York VAN DEN BERG, J 00 Bivariate Distributions with Correlated Chi-square Marginals: Product and Ratio Coponents Msc thesis, University of South Africa UNISA VAN DEN BERG, J, ROUX, J J J, AND BEKKER, A 03 A bivariate generalization of gaa distribution Counications in Statistics - Theory and Methods, 4 9, YUNUS, R M AND KHAN, S 0 The bivariate noncentral chi-square distribution a copound distribution approach Applied Matheatics and Coputation, 7, Manuscript received, , revised, , accepted,