ARNOLD AND STRAUSS S BIVARIATE EXPONENTIAL DISTRIBUTION PRODUCTS AND RATIOS. Saralees Nadarajah and Dongseok Choi (Received February 2005)

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1 NEW ZEALAND JOURNAL OF MATHEMATICS Volume 35 6), ARNOLD AND STRAUSS S BIVARIATE EXPONENTIAL DISTRIBUTION PRODUCTS AND RATIOS Saralees Nadarajah and Dongseok Choi Received February 5) Abstract. We derive the distributions of W = X/X + Y ) equivalently, X/Y ) and P = XY and the the corresponding moment properties when X and Y follow Arnold and Strauss s bivariate exponential distribution. The expressions turn out to involve several special functions. We also provide extensive tabulations of the percentage points associated with the two distributions. These tables obtained using intensive computing power will be of use to practitioners of the bivariate exponential distribution. 1. Introduction For a bivariate random vector X, Y ), the distributions of the ratios X/X + Y ) and X/Y and the product XY are of interest in problems in biological and physical sciences, econometrics, classification, and ranking and selection. 1) Examples of the use of the ratio of random variables include Mendelian inheritance ratios in genetics, mass to energy ratios in nuclear physics, target to control precipitation in meteorology, and inventory ratios in economics. The distributions of X/X + Y ) and X/Y have been studied by several authors especially when X and Y are independent random variables and come from the same family. For instance, see Marsaglia 1965) and Korhonen and Narula 1989) for normal family, Press 1969) for Student s t family, Basu and Lochner 1971) for Weibull family, Shcolnick 1985) for stable family, Hawkins and Han 1986) for non central chi squared family, Provost 1989b) for gamma family, and Pham Gia ) for beta family. ) As an example of the use of the product of random variables in Physics, Sornette 1998) mentions:... To mimic system size limitation, Takayasu, Sato, and Takayasu introduced a threshold x c... and found a stretched exponential truncating the power law pdf beyond x c. Frisch and Sornette recently developed a theory of extreme deviations generalizing the central limit theorem which, when applied to multiplication of random variables, predicts the generic presence of stretched exponential pdfs. The problem thus boils down to determining the tail of the pdf for a product of random variables... The distribution of XY has been studied by several authors especially when X and Y are independent random variables and come from the same family. For instance, see Sakamoto 1943) for uniform family, Harter 1951) and 1991 Mathematics Subject Classification 33C9, 6E99.

2 19 SARALEES NADARAJAH AND DONGSEOK CHOI Wallgren 198) for Student s t family, Springer and Thompson 197) for normal family, Stuart 196) and Podolski 197) for gamma family, Steece 1976), Bhargava and Khatri 1981) and Tang and Gupta 1984) for beta family, AbuSalih 1983) for power function family, and Malik and Trudel 1986) for exponential family see also Rathie and Rohrer 1987) for a comprehensive review of known results ). There is relatively little work of the above kind when X and Y are correlated random variables. Some of the known work for ratios include Hinkley 1969) for bivariate normal family, Kappenman 1971) for bivariate t family, and Lee et al 1979) for bivariate gamma family. The only work known to the author for products is that by Garg et al ) for Dirichlet family.in this paper, we consider the distributions of W = X/X + Y ) equivalently, X/Y ) and P = XY when X and Y are correlated exponential random variables with the joint pdf f x, y) = K exp { ax + by + cxy)} 1) for x >, y >, a >, b > and c >, where K = Ka, b, c) is the normalizing constant given by ) 1 ab = c exp Ei ab ) K c c and Ei ) is denotes the exponential integral function defined by Ei x) = x expt) dt. t This distribution is due to Arnold and Strauss 1988) and is known as the conditionally specified bivariate exponential distribution. The marginal pdf of X and the conditional pdf of X given Y = y are and f X x) = K exp ax) b + cx f X Y x y) = a + cy) exp { a + cy)x}, respectively. As often with the exponential distribution, 1) has applications in reliability studies. Inaba and Shirahata 1986) fitted 1) to data on white blood counts and survival times of patients who died of acute myelogenous leukemia Gross and Clark, 1975), comparing it with the bivariate normal distribution. Furthermore, note that 1) belongs to the exponential family. Thus, by Lemma 8 in Lehmann 1997), one can obtain confidence intervals for a, b and c by conditioning on part of the sufficient statistic when sampling from 1). The paper is organized as follows. In Sections and 3, we derive explicit expressions for the pdfs, cdfs and moments of W = X/X + Y ) equivalently, X/Y ) and P = XY. In Section 4, we provide extensive tabulations of the associated percentage points, obtained by means of intensive computing power. These values will be of use to the practitioners of the bivariate gamma distribution.

3 PRODUCTS AND RATIOS 191 The calculations of this paper involve several special functions, including the modified Bessel function of the third kind defined by πx ν K ν x) = ν exp xt) t 1 ) ν 1/ dt, Γ ν + 1/) and, the Kummer function defined by Ψa, b; x) = 1 Γa) We also need the following important lemmas. t a t) b a 1 exp xt)dt. Lemma 1.1 Equation ), Prudnikov et al., 1986, volume 1). For p >, x n exp px qx ) π dx = 1) n n { ) q p q n exp Φ q )}, 4p p where Φ ) denotes the cumulative distribution function of the standard normal distribution. Lemma 1. Equation ), Prudnikov et al., 1986, volume 1). For p > and q >, x α 1 exp px q/x) dx = q/p) α/ K α pq). Lemma 1.3 Equation.3.6.9), Prudnikov et al., 1986, volume 1). For α > and p >, x α 1 exp px) x + z) ρ dx = Γα)z α ρ Ψ α, α + 1 ρ; pz). The properties of the above special functions can be found in Prudnikov et al. 1986) and Gradshteyn and Ryzhik ).. and CDF Theorems.1 to. derive the pdfs of W = X/X + Y ) and P = XY when X and Y are distributed according to 1). The corresponding cdf for R = X/Y is also given in Theorem.1. Theorem.1. If X and Y are jointly distributed according to 1) then the pdf of W = X/X + Y ) is given by { } K ΦC) f W w) = 4c A φc) ca ) for < w < 1, where A = w1 w), B = aw + b1 w), C = B/{ ca} and φ ) denotes the pdf of the standard normal distribution. Equivalently, the pdf of R = X/Y is given by { } K ΦC) f R r) = 4c A 1 + r) φc) ca 3)

4 19 SARALEES NADARAJAH AND DONGSEOK CHOI for < r <, where A = r/{1 + r) }, B = ar + b)/1 + r) and C = B/{ ca}. Furthermore, the cdf of R = X/Y can be expressed in the series form F R r) = 1 K { ar + b) } a cr exp c m 4cr a) m ) m n m ar + b) 1) n n cr) h m n/ n where h n x) is given by m= h n x) = x n= z n exp ) z dz. Proof. From 1), the joint pdf of S, W ) = X + Y, X/S) becomes fs, w) = Ks exp [ s {aw + b1 w)} cs w1 w) ]. Thus, the pdf of W can be written as f W w) = K s exp [ s {aw + b1 w)} cs w1 w) ] ds. ) ar + b, cr The result in ) follows by using Lemma 1.1 to calculate the above integral. The result in 3) follows by noting that if f W ) denotes the pdf of W then that of R is given 1 + r) f W r/1 + r)). The cdf of R can be calculated as follows: P R < r) = 1 P X > ry ) = 1 K ry exp { ax + by + cxy)} dxdy exp { ar + b)y cry } = 1 K dy a + cy { ar + b) } { 1 = 1 K exp 4cr a + cy exp cr y + ar + b ) } dy cr = 1 K { ar + b) } a exp 1) m c ) m { y m exp cr 4cr a m= = 1 K { ar + b) } a cr exp 1) m c ) m 4cr a m= { z ar+b)/ ar + b } m ) exp z dz cr cr cr = 1 K { ar + b) } a cr exp 1) m c ) m n ) m ar + b 4cr a n cr m= n= ar+b)/ z n exp z cr 4) y + ar + b ) } dy cr ) m n 1 cr ) n ) dz, 5) where we have substituted z = cr{y + ar + b)/cr)}. The result in 4) follows immediately from 5).

5 PRODUCTS AND RATIOS 193 Checking convergence of the infinite series in 4) is an open problem. Note also that the terms h n ) have been widely used elsewhere in statistics and recursive formulas are available for their computation. Theorem.. If X and Y are jointly distributed according to 1) then f P p) = K exp cp)k ) abp for < p <. Proof. From 1), the joint pdf of X, P ) = X, XY ) becomes fx, p) = Kx 1 exp cp) exp ax bp ). x Thus, the pdf of P can be written as f P p) = Kp β 1 exp cp) 6) x 1 exp ax bp ) dx. 7) x The result of the theorem follows by using Lemma 1. to calculate the above integral. a) b) w w c) d) w w Figure 1. Plots of the pdf of ) for a): c =.1; b): c = 1; c): c = 3; and, d): c = 5. The four curves in each plot are: the solid curve a = 1, b = 1), the curve

6 194 SARALEES NADARAJAH AND DONGSEOK CHOI of lines a = 1, b = 3), the curve of dots a = 3, b = 1), and the curve of lines and dots a = 3, b = 3). a) b) p p c) d) p p Figure. Plots of the pdf of 6) for a): c =.1; b): c = 1; c): c = 3; and, d): c = 5. The four curves in each plot are: the solid curve a = 1, b = 1), the curve of lines a = 1, b = 3), the curve of dots a = 3, b = 1), and the curve of lines and dots a = 3, b = 3). Figures 1 to illustrate the shape of the pdfs ) and 6) for selected values of a, b and c. Each plot contains four curves corresponding to selected values of a and b. The effect of the parameters is evident. 3. Moments Here, we derive the moments of P = XY when X and Y are distributed according to 1). We need the following lemma. Lemma 3.1. If X and Y are jointly distributed according to 1) then m!n!b m n c m Ψ m + 1, m n + 1; ab ) E X m Y n c ) = Ψ 1, 1; ab ) c for m 1 and n 1.

7 PRODUCTS AND RATIOS 195 Proof. One can write E X m Y n ) = K = K = K = Kn! x m y n fx, y)dydx, x m y n exp { ax + by + cxy)} dydx x m exp ax) = Km!n!b m n c m+1) Ψ y n exp { b + cx)y} dydx x m exp ax)b + cx) n+1) dx m + 1, m n + 1; ab ), c where the last equality follows by application of Lemma 1.3. The normalizing constant K is the reciprocal of same integral above when m = and n =. The result of the lemma is immediate. The moments of P = XY are now an immediate consequence of this lemma. Theorem 3.. If X and Y are jointly distributed according to 1) then n!n!c n Ψ n + 1, 1; ab ) E P n c ) = Ψ 1, 1; ab ) c for n 1. Proof. Follows by writing EP n ) = EX n Y n ) and applying the lemma with m = n. 4. Percentiles In this section, we provide extensive tabulations of the percentiles of the distributions of W and P. These percentiles are computed numerically by solving the equations and wα pα f W w)dw = α f P p)dp = α, where f W w) and f P p) are given by ) and 6), respectively. Evidently, this involves computation of the complementary error and Bessel functions and routines for this are widely available. We used the functions erfc ) and BesselK ) in the algebraic manipulation package, MAPLE. The percentiles are given for α =.9,.95,.975,.99,.995,.999,.9995, a = 1, b = 1 and c =.1,.,..., 5. c Percentage points, w α, for W = X/X + Y ) α

8 196 SARALEES NADARAJAH AND DONGSEOK CHOI

9 PRODUCTS AND RATIOS 197 Percentage points, p α, for P = XY c α

10 198 SARALEES NADARAJAH AND DONGSEOK CHOI Similar tabulations could be easily derived for other values of a, b and c. We hope these numbers will be of use to the practitioners of the bivariate exponential distribution see Section 1). Acknowledgements. The authors would like to thank the referee and the editor for carefully reading the paper and for their great help in improving the paper. References 1. M.S. Abu-Salih, Distributions of the product and the quotient of power-function random variables, Arab Journal of Mathematics, ), B.C. Arnold and D.J. Strauss, Bivariate distributions with exponential conditionals, Journal of the American Statistical Association, ), A.P. Basu and R.H. Lochner, On the distribution of the having generalized life distributions, Technometrics, ), R.P. Bhargava and C.G. Khatri, The distribution of product of independent beta random variables with application to multivariate analysis, Annals of the Institute of Statistical Mathematics, ), M. Garg, V. Katta and M.K. Gupta, The distribution of the products of powers of generalized Dirichlet components, Kyungpook Mathematical Journal, 4 ), I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, sixth edition), San Diego, Academic Press,. 7. A.J. Gross and V.A. Clark, Survival Distributions: Reliability Applications in the Biomedical Sciences, New York, John Wiley and Sons, H.L. Harter, On the distribution of Wald s classification statistic, Annals of Mathematical Statistics, 1951), D.L. Hawkins and C.-P. Han, Bivariate distributions noncentral chi square random variables, Communications in Statistics Theory and Methods, ), D.V. Hinkley, On the ratio of two correlated normal random variables, Biometrika, ), T. Inaba and S. Shirahata, Measures of dependence in normal models and exponential models by information gain, Biometrika, ), R.F. Kappenman, A note on the multivariate t ratio distribution, Annals of Mathematical Statistics, ), P.J. Korhonen and S.C. Narula, The probability distribution of the ratio of the absolute values of two normal variables, Journal of Statistical Computation and Simulation, ), R.Y. Lee, B.S. Holland and J.A. Flueck, Distribution of a ratio of correlated gamma random variables, SIAM Journal on Applied Mathematics, ), E.L. Lehmann, Testing Statistical Hypotheses, New York, Springer Verlag, 1997.

11 PRODUCTS AND RATIOS H.J. Malik and R. Trudel, Probability density function of the product and quotient of two correlated exponential random variables, Canadian Mathematical Bulletin, ), G. Marsaglia, Ratios of normal variables and ratios of sums of uniform variables, Journal of the American Statistical Association, ), T. Pham Gia, Distributions of the ratios of independent beta variables and applications, Communications in Statistics Theory and Methods, 9 ), H. Podolski, The distribution of a product of n independent random variables with generalized gamma distribution, Demonstratio Mathematica, 4 197), S.J. Press, The t ratio distribution, Journal of the American Statistical Association, ), S.B. Provost, On the distribution of the ratio of powers of sums of gamma random variables, Pakistan Journal Statistics, ), A.P. Prudnikov, Y.A. Brychkov and O.I. Marichev, Integrals and Series, Volumes 1, and 3), Amsterdam, Gordon and Breach Science Publishers, P.N. Rathie and H.G. Rohrer, The exact distribution of products of independent random variables, Metron, ), H. Sakamoto, On the distributions of the product and the quotient of the independent and uniformly distributed random variables, Tohoku Mathematical Journal, ), S.M. Shcolnick, On the ratio of independent stable random variables, Stability Problems for Stochastic Models, Uzhgorod, 1984, pp , Lecture Notes in Mathematics, Vol. 1155, Springer, Berlin, D. Sornette, Multiplicative processes and power laws, Physical Review E, ), M.D. Springer and W.E. Thompson, The distribution of products of beta, gamma and Gaussian random variables, SIAM Journal on Applied Mathematics, ), B.M. Steece, On the exact distribution for the product of two independent beta distributed random variables, Metron, ), A. Stuart, Gamma distributed products of independent random variables, Biometrika, ), J. Tang and A.K. Gupta, On the distribution of the product of independent beta random variables, Statistics and Probability Letters, 1984), C.M. Wallgren, The distribution of the product of two correlated t variates, Journal of the American Statistical Association, ), Saralees Nadarajah School of Mathematics University of Manchester Oxford Road Manchester M13 9PL UNITED KINGDOM Saralees.Nadarajah@manchester.ac.uk Dongseok Choi Department of Public Health and Preventive Medicine Oregon Health and Science University Portland Oregon 9739 USA choid@ohsu.edu