Moments of the product and ratio of two correlated chi-square variables

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1 Stat Papers : DOI /s REGULAR ARTICLE Moents of the product and ratio of two correlated chi-square variables Anwar H. Joarder Received: June 006 / Revised: 8 October 007 / Published online: 0 Noveber 007 The Authors 007 Abstract The exact probability density function of a bivariate chi-square distribution with two correlated coponents is derived. Soe oents of the product and ratio of two correlated chi-square rando variables have been derived. The ratio of the two correlated chi-square variables is used to copare variability. One such application is referred to. Another application is pinpointed in connection with the distribution of correlation coefficient based on a bivariate t distribution. Keywords Bivariate chi-square distribution Moents Product of correlated chi-square variables Ratio of correlated chi-square variables Matheatics Subject Classification 000 6E15 60E05 60E10 1 Introduction Fisher 1915 derived the distribution of ean-centered su of squares and su of products in order to study the distribution of correlation coefficient fro a bivariate noral saple. Let X 1, X,...,X N N > be two-diensional independent rando vectors where X j = X 1 j, X j, j = 1,,...,N is distributed as a bivariate noral distribution denoted by N θ, with θ = θ 1,θ and a covariance atrix = σ ik, i = 1, ; k = 1,. The saple ean-centered sus of squares and su of products are given by a ii = N j=1 X ij X i = Si, = N 1,i = 1, and a 1 = N j=1 X 1 j X 1 X j X = RS 1 S, respectively. The quantity R A. H. Joarder B Departent of Matheatics and Statistics, King Fahd University of Petroleu and Minerals, Dhahran 3161, Saudi Arabia e-ail: anwarj@kfup.edu.sa; ajstat@gail.co 13

2 58 A. H. Joarder is the saple product oent correlation coefficient. The distribution of a 11, a and a 1 was derived by Fisher 1915 and ay be called the bivariate Wishart distribution after Wishart 198 who obtained the distribution of a p-variate Wishart atrix as the joint distribution of saple variances and covariances fro the ultivariate noral population. Obviously a 11 /σ 11 has a chi-square distribution with degrees of freedo. Let U 1 = S 1 /σ 1 and U = S /σ. The joint distribution of U 1 and U is called the bivariate chi-square distribution after Krishnaiah et al There are also a nuber of other bivariate chi-square and gaa distributions excellently reviewed by Kotz et al In this paper we derive its exact probability density function pdf in Theore.1. Soe properties of the distribution available in literature are reviewed in Kotz et al We feel the properties need to be thoroughly investigated in the light of the joint pdf derived in this paper. We provide oents of the product of two correlated chi-square rando variables in Corollaries 3. and 3.3. In case the correlation coefficient vanishes, the oents, as expected, coincide with that of the bivariate independent chi-squares rando variables. An application is pinpointed in connection with the distribution of saple product oent correlation coefficient based on a bivariate t distribution. The bivariate t distribution has fatter tails copared to bivariate noral distribution and has found nuerous applications to business data especially stock return data. Ratios of two independent chi-squares are widely used in statistical tests of hypotheses. However, testing with the ratio of correlated chi-squares variables are rarely noticed. If σ 1 = σ, Finney 1938 derived the sapling distribution of the square root of the ratio of correlated chi-squares variables T = U/V directly fro the joint distribution of saple variances and correlation coefficient. He copared the variability of the easureents of standing height and ste length for different age groups of schoolboys by his ethod and by Hirsfeld In Corollaries 3.7 and 3.8, we have provided oents of the ratio of two correlated chi-square variables. Further investigation is needed to utilize these oents to derive distributional properties of correlated chi-square variables by the inverse Mellin transfor Provost It needs further investigation to derive the arginal and conditional distribution, and the distribution of the su, product, ratio and siilar quantities directly fro the joint density function of the bivariate chi-square distribution with pdf in Theore.1. In what follows, for any nonnegative integer k, we will use the notation c k = cc + 1 c + k 1, c 0 = 1, c = 0. The pdf of the bivariate chi-square distribution The pdf of the bivariate Wishart distribution originally derived by Fisher 1915 can be written as 13

3 Moents of the product and ratio of two correlated chi-square variables 583 Fig. 1 Correlated bivariate chi-square density f x, y surface with ρ = 0 versus independent bivariate chi-square density gx, y surfaces at 3 a, b, 6 c, d,and9e, f degrees of freedo f 1 a 11, a, a 1 = 1 ρ / σ 1 σ π 1 a 11 a a1 3/ a 11 a ρa 1 exp 1 ρ σ1 1 ρ σ + 1 ρ σ 1 σ.1 where a 11 > 0, a > 0, a 11 a < a 1 < a a, >,σ 1 = σ 11, σ = σ,σ 1 = ρσ 1 σ,σ 1 > 0, σ > 0 and ρ, 1 <ρ<1, is the product oent correlation coefficient between X 1 j and X j j = 1,,...,N Anderson 003, p. 13 Figs. 1,. Since the correlation coefficient between X 1 j and X j j = 1,,...,N isρ, the correlation between U 1 and U is ρ Joarder 006. In the following theore we derive the pdf of the correlated bivariate chi-square distribution. 13

4 584 A. H. Joarder Fig. Correlated bivariate chi-square density f x, y surface with 3 degrees of freedo at different values of ρ: a ρ = 0.99, b ρ = 0.99, c ρ = 0.5, d ρ = 0.5, e ρ = 0.85, and f ρ = 0.85 Theore.1 The rando variables U 1 and U are said to have a correlated bivariate chi-square distribution each with degrees of freedo, if its pdf is given by f u 1, u = +1 u 1 u / u 1 +u e 1 ρ π 1 ρ / = N 1 >, 1 <ρ<1. ρ [1+ 1 k u1 u k k+1 1 ρ k!, k+ Proof Under the transforation a 11 = s1, a = s, a 1 = rs 1 s in.1 with Jacobian Ja 11, a, a 1 s1, s, r = 3 s 1 s, the pdf of S1, S and R is given by 13 f s 1, s, r = 1 ρ / π 1 1 r 3/ s 1 s σ 1 σ s1 s exp 1 ρ σ1 1 ρ σ + ρrs 1s 1 ρ. σ 1 σ

5 Moents of the product and ratio of two correlated chi-square variables 585 By aking the transforation s 1 = σ 1 u 1, s = σ u, keeping r intact, with Jacobian J s 1, s u 1u = σ1 σ /,wehave f 3 u 1, u, r = 1 ρ / u 1 u / 1 r 3/ π 1 exp u 1 + u 1 ρ + ρr u1 u 1 ρ. Then integrating out r, the joint pdf of U 1 and U is given by f 3 u 1, u = 1 ρ / u 1 u / π exp u 1 + u 1 ρ 1 r 3/ ρr u1 u exp 1 ρ dr. By expanding the last exponential ter of the above density function we have f 3 u 1, u = 1 ρ / u 1 u / π 1 k 1 k! which can be written as ρr u1 u 1 ρ exp 0 1 f 3 u 1, u = 1 ρ / u 1 u / π 1 u 1 + u 1 ρ r k 1 r 3/ + exp ρ [ 1 k u1 u y k ρ k! u 1 + u 1 ρ r k 1 r 3/ dr 1 y 3/ 1 y 1/ dy. By copleting the beta integral of the above density function, we have f 3 u 1, u = 1 ρ / u 1 u / +1 π 1 ρ [1 + 1 k u1 u y 1 ρ which siplifies to what we have the theore. exp k 1 k! u 1 + u 1 ρ k+1 1 k+ 13

6 586 A. H. Joarder Note that the pdf of the correlated bivariate chi-square distribution each with degrees of freedo can be written as f u 1, u = u 1u / u 1 +u e 1 ρ π 1 ρ / ρ u1 u 1 ρ k k+1 k!, k+ = N 1 >, 1 <ρ<1, and k = 0,, 4,... In case ρ = 0, the pdf in Theore.1 becoes the product of that of the two independent chi-square rando variables each with degrees of freedo. 3 Moents of the product and ratio of two correlated chi-square rando variables Theore 3.1 The a, b-th product oent µ a, b; ρ = E U a 1 U b is given by E U1 a U b = a+b 1 1 ρ a+b L,ρ [1 + 1 k ρ k k + k! k+ k + + a + b k + 1 where > axa, b, 1 <ρ<1and L,ρ= π/ 1 ρ /. 3.1 Proof It follows fro Theore.1 that the a, b-th product oent of the distribution of U 1 and U is given by E U1 a U b = 1 ρ / π ρ k [1 + 1 k 1 1 ρ k! 0 0 k+1 k+ u +k +a 1 1 u +k +b 1 e u1+u 1 ρ du 1 du. The theore then follows by having evaluated the above integrals. Obviously µ a, b; ρ = µ b, a; ρ = EU1 au b.ifρ = 0, the product oent in Theore 3.1 siplifies to 13 EU a 1 U b = a+b / + a + b

7 Moents of the product and ratio of two correlated chi-square variables 587 which is the product of the a-th and b-th oents of two independent variables U 1 χ and U χ. In case a is a nonnegative integer, then k + k + k + + a =. 3. a Corollary 3.1 If the integers i a and b are nonnegative, or, ii a > 0, b < 0 with > b, then the product oent in Theore 3.1 can be written as E U1 a U b = a+b 1 ρ a+b+/ π k + ρ k γ k,+b a 3.3 where γ k, =[1 + 1 k k 1 k! k + 1 k Plugging in a = b in Corollary 3.1, we have the following corollary. Corollary 3. For >, 1 <ρ<1, the a-th oent of V = U 1 U is given by EV a = 4a 1 ρ a+/ π where γ k, is defined by 3.4. k+ + a k+ γ k,+a ρ k 3.5 In case a is a nonnegative integer, then by the use of 3. in3.5 wehavethe following: E V a = 4a 1 ρ a+/ π k + γ k,+a ρ k 3.6 a where γ k, is defined by 3.4. The oent generating function of V at h is given by M V h = E e hv = E a=0 hv a a! = a=0 h a a! EV a. The oents of V are evaluated fro Corollary 3.. Corollary 3.3 For >, 1 <ρ<1, the first four raw oents of V are given by i EV = + ρ, ii EV = + [ 8ρ ρ + +, iii EV 3 = 16 [ 3 4ρ ρ ρ + 4 3, 13

8 588 A. H. Joarder iv EV 4 = 56 4[ 4ρ ρ ρ ρ + 4. Proof Define L r,ρ = k r γ k, ρ k, with L 0,ρ = γ k, ρ k which is denoted by L,ρJoarder 006, and given in 3.1 in this paper. Then it can be checked that L 1 +,ρ= + ρ 1 ρ L +,ρ, and that ρ k γ k,+ k + / = 1 [ L 1 +,ρ+ L +,ρ = 1 [ + ρ 1 ρ 1 + L +,ρ π [ = ρ 1 ρ ρ +/. By plugging the above in to 3.6 with a = 1, we have EV = 1 ρ [ + + ρ 1 ρ which siplifies to i. ii Using the following identities L 1 + 4,ρ = + 4ρ 1 ρ L + 4,ρ and L + 4,ρ =[ ρ ρ 1 ρ L + 4,ρ in the expression ρ k γ k,+4 k + / = 1 4 [ L + 4,ρ + + 3L 1 + 4,ρ+ + L + 4,ρ and plugging in the value of L + 4,ρin 3.6 with a =, we get ii. Siilarly iii and iv can be proved. In has been entioned at the end of Theore.1 that if ρ = 0, then U 1 χ and U χ will be independent. In that case, V will be the product of two independent chi-square rando variables each with degrees of freedo and evidently the resulting oents will be in agreeent with that situation. For exaple if ρ = 0, fro Corollary 3.3 iv we have EV 4 =[ which is the product of the fourth oents of two independent chi-squares each having degrees of freedo, i.e., EV 4 = EU1 4EU 4. Since the geoetric ean of the chi-square variables are iportant in any applications, we define G = U 1 U, the geoetric ean of U 1 and U, and provide its oents below. 13

9 Moents of the product and ratio of two correlated chi-square variables 589 Corollary 3.4 For >, 1 <ρ<1, the first four raw oents of G are given by i EG = 1 ρ +/ π ii E G = + ρ, k++1 k+ k++3 k+ γ k,+1 ρ k, iii E G 3 = 81 ρ +6/ γ k,+3 ρ k, π iv EG 4 = + [ 8ρ ρ + +. Since G = U 1 U = V, the oent generating function of G at h is given by Ee hg = h a a=0 a! EGa = h a a=0 a! EV a/. Then we have the following corollary. Corollary 3.5 If a is nonnegative with > a, then for 1 <ρ<1, the oent generating function of G at h is given by M G h = 1 ρ / π h a 1 ρ a a=0 a! k++a γ k+ k,+a ρ k, where γ k, is defined by 3.4. The following results fro Corollaries 3. and 3.5 by putting ρ = 0. Corollary 3.6 Let G = U 1 U = V be the geoetric ean of two independent chi-square rando variables U 1 and U each with degrees of freedo. Then the a-th oent and the oent generating function of G at h are given by EG a = a +a M G h = h a a=0 a! and +a, respectively Joarder 007. Let W = U 1 /U, the ratio of two correlated chi-square variables U 1 and U that have pdf in Theore.1. Then the following result follows fro Theore 3.1. Corollary 3.7 If a is nonnegative with > a, then for 1 <ρ<1, thea-th oent of W is given by EW a = 1 ρ / π/ k+ + a k+ γ k, a ρ k, where γ k, a is defined by 3.4. In case a is a nonnegative integer, we have EW a = 1 ρ / π k+ a γ k, a ρ k, > a, 1 <ρ<1. 13

10 590 A. H. Joarder The oents of W up to fourth order are calculated below fro Corollary 3.7. The derivation is siilar to that of Corollary 3.3. Corollary 3.8 For > 8, 1 <ρ<1, the first four oents of W are given below: i EW = ρ, >, ii E W 1 = 4 4ρ ρ + +, > 4, 3 iii EW 3 = [ 480ρ ρ ρ + 8 iv 8 > 6, 4 EW 4 = [ 13440ρ ρ ρ ρ + 8, > 8. 3 In case ρ = 0, then W will be the ratio of two independent chi-square variables each with the sae degrees of freedo, and the above oents will be siply EW a = a a, > a a 3, which are evidently in agreeent with the resulting situation of independence. Corollary 3.9 If a is an integer with > a, then for 1 <ρ<1, the oent generating function of W at h is given by M W h = 1 ρ / π a=0 where L,ρis defined in Theore 3.1. h a a! k + γ k, a ρ k a 4 The distribution of the correlation coefficient based on bivariate t population The lengthy proof of the distribution of saple correlation coefficient based on a bivariate noral distribution by Fisher 1915 has been ade shorter and elegant by Joarder 007. Further he derived the distribution of correlation coefficient for bivariate t distribution along Fisher It is known that the distribution of correlation coefficient is robust under violation of norality at least in the bivariate elliptical class of distributions Fang and Anderson 1990, p. 10; Fang and Zhang 1990, p. 137, or Ali and Joarder However a direct derivation by Joarder 007 provides ore insight for those who have been recently using t distributions as parent populations. 13

11 Moents of the product and ratio of two correlated chi-square variables 591 Theore 4.1 The pdf of the correlation coefficient R based on a bivariate t population is given by i hr = 1 ρ / π 1 1 r 3/ M G ρr ii hr = 1 ρ / π 1 1 r 3/ ρr k k! +k, 1 < r < 1cf. Muirhead 198, p. 154 where >, 1 <ρ<1and M G ρr is defined in Corollary 3.6. It is overviewed here to show the application of the oents of Sect. 3. Since part ii of Theore 4.1 is well known to be the distribution of the correlation coefficient for bivariate noral population, it proves the robustness of the distribution or of tests on correlation coefficient. Thus the assuption of bivariate norality under which any tests on correlation coefficient are developed can be relaxed to a broader class of bivariate t distribution. Interested readers ay go through Muddapur 1988 for soe exciting exact tests on correlation coefficient. Further the readers are referred to Joarder and Ahed 1996, Billah and Saleh 000, Kibria 004, Kibria and Saleh 004, Kotz and Nadarajah 004 and the references therein for applications of t distribution. Acknowledgents The author acknowledges the excellent research support provided by King Fahd University of Petroleu and Minerals, Saudi Arabia through the Project FT 004. The author also thanks the referees, his colleague Dr. M.H. Oar, and Professor M. Isail, University of Central Florida, for constructive suggestions that have iproved the original version in any ways. Open Access This article is distributed under the ters of the Creative Coons Attribution Noncoercial License which perits any noncoercial use, distribution, and reproduction in any ediu, provided the original authors and source are credited. References Ali MM, Joarder AH 1991 Distribution of the correlation coefficient for the class of bivariate elliptical odels. Can J Stat 19: Anderson TW 003 An introduction to ultivariate statistical analysis. Wiley, London Billah MB, Saleh AKME 000 Perforance of the large saple tests in the actual forecasting of the pretest estiators for regression odel with t error. J Appl Stat Sci 93:37 51 Fang KT, Anderson TW 1990 Statistical inference in elliptically contoured and related distributions. Allerton Press Fang KT, Zhang YT 1990 Generalized ultivariate analysis. Springer, Heidelberg Finney DJ 1938 The distribution of the ratio of the estiates of the two variances in a saple fro a bivariate noral population. Bioetrika 30: Fisher RA 1915 Frequency distribution of the values of the correlation coefficient in saples fro an indefinitely large population. Bioetrika 10: Hirsfeld HO 1937 The distribution of the ratio of covariance estiates in two saples drawn fro noral bivariate populations. Bioetrika 9:65 79 Joarder AH 006 Product oents of bivariate Wishart distribution. J Probab Stat Sci 4:33 43 Joarder AH 007 Soe useful integrals and their applications in correlation analysis. Stat Papers in press Joarder AH, Ahed SE 1996 Estiation of characteristic roots of scale atrix. Metrika 4:59 67 Kibria BMG 004 Conflict aong the shrinkage estiators induced by W, LR and LM tests under a Student s t regression odel. J Korean Stat Soc 334: Kibria BMG, Saleh AKME 004 Preliinary test ridge regression estiators with Students t errors and conflicting test statistics. Metrika 59:

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