ARNOLD AND STRAUSS’S BIVARIATE EXPONENTIAL DISTRIBUTION – PRODUCTS AND RATIOS
Saralees Nadarajah and Dongseok Choi
(Received February 2005)
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 expres- sions 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 inher- itance 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 (2000) for beta family.
(2) As an example of the use of the product of random variables in Physics, Sor- nette (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 re- cently 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 expo- nential 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 fam- ily. For instance, see Sakamoto (1943) for uniform family, Harter (1951) and
1991 Mathematics Subject Classification 33C90, 62E99.
Wallgren (1980) for Student’s t family, Springer and Thompson (1970) for normal family, Stuart (1962) and Podolski (1972) for gamma family, Steece (1976), Bhargava and Khatri (1981) and Tang and Gupta (1984) for beta fam- ily, 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 (2002) 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 > 0, y > 0, a > 0, b > 0 and c > 0, where K = K(a, b, c) is the normalizing constant given by
1
K = −c exp ab c
Ei
− ab c
and Ei(·) is denotes the exponential integral function defined by Ei (x) =
Z x
−∞
exp(t) t dt.
This distribution is due to Arnold and Strauss (1988) and is known as the condi- tionally specified bivariate exponential distribution. The marginal pdf of X and the conditional pdf of X given Y = y are
f X (x) = K exp(−ax) b + cx and
f X|Y (x|y) = (a + cy) exp {−(a + cy)x} ,
respectively. As often with the exponential distribution, (1) has applications in reli- ability 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 2 and 3, we derive explicit ex-
pressions 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.
The calculations of this paper involve several special functions, including the modified Bessel function of the third kind defined by
K ν (x) =
√ πx ν 2 ν Γ (ν + 1/2)
Z ∞ 0
exp(−xt) t 2 − 1 ν−1/2
dt, and, the Kummer function defined by
Ψ(a, b; x) = 1 Γ(a)
Z ∞ 0
t a−1 (1 + t) b−a−1 exp(−xt)dt.
We also need the following important lemmas.
Lemma 1.1 (Equation (2.3.15.7), Prudnikov et al., 1986, volume 1). For p > 0, Z ∞
0
x n exp −px 2 − qx dx = (−1) n r π p
∂ n
∂q n
exp q 2 4p
Φ
− q
√ p
, where Φ(·) denotes the cumulative distribution function of the standard normal distribution.
Lemma 1.2 (Equation (2.3.16.1), Prudnikov et al., 1986, volume 1). For p > 0 and q > 0,
Z ∞ 0
x α−1 exp (−px − q/x) dx = 2(q/p) α/2 K α (2 √ pq) .
Lemma 1.3 (Equation (2.3.6.9), Prudnikov et al., 1986, volume 1). For α > 0 and p > 0,
Z ∞ 0
x α−1 exp(−px)
(x + z) ρ dx = Γ(α)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 (2000).
2. PDF and CDF
Theorems 2.1 to 2.2 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 2.1.
Theorem 2.1. If X and Y are jointly distributed according to (1) then the pdf of W = X/(X + Y ) is given by
f W (w) = K 4c 2 A 2
Φ(C) φ(C) − 2cA
(2) for 0 < w < 1, where A = w(1 − w), B = aw + b(1 − w), C = B/{2 √
cA} and φ(·) denotes the pdf of the standard normal distribution. Equivalently, the pdf of R = X/Y is given by
f R (r) = K
4c 2 A 2 (1 + r) 2
Φ(C) φ(C) − 2cA
(3)
for 0 < r < ∞, where A = r/{(1 + r) 2 }, B = (ar + b)/(1 + r) and C = B/{2 √ cA}.
Furthermore, the cdf of R = X/Y can be expressed in the series form F R (r)
= 1− K
a √
2cr exp (ar + b) 2 4cr
∞ X
m=0
c a
m X m
n=0
(−1) n m n
(ar + b) m−n (2cr) m−n/2 h n
ar + b
√ 2cr
, (4) where h n (x) is given by
h n (x) = Z ∞
x
z n exp
− z 2 2
dz.
Proof. From (1), the joint pdf of (S, W ) = (X + Y, X/S) becomes f (s, w) = Ks exp −s {aw + b(1 − w)} − cs 2 w(1 − w) . Thus, the pdf of W can be written as
f W (w) = K Z ∞
0
s exp −s {aw + b(1 − w)} − cs 2 w(1 − w) ds.
The result in (2) 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) −2 f W (r/(1 + r)). The cdf of R can be calculated as follows:
P (R < r)
= 1 − P (X > rY )
= 1 − K Z ∞
0
Z ∞ ry
exp {− (ax + by + cxy)} dxdy
= 1 − K Z ∞
0
exp −(ar + b)y − cry 2
a + cy dy
= 1 − K exp (ar + b) 2 4cr
Z ∞ 0
1 a + cy exp
(
−cr
y + ar + b 2cr
2 ) dy
= 1 − K
a exp (ar + b) 2 4cr
∞ X
m=0
(−1) m c a
m Z ∞ 0
y m exp (
−cr
y + ar + b 2cr
2 ) dy
= 1 − K a √
2cr exp (ar + b) 2 4cr
∞ X
m=0
(−1) m c a
m
× Z ∞
(ar+b)/ √ 2cr
z
√
2cr − ar + b 2cr
m exp
− z 2 2
dz
= 1 − K a √
2cr exp (ar + b) 2 4cr
∞ X
m=0
(−1) m c a
m X n
n=0
m n
− ar + b 2cr
m−n 1
√ 2cr
n
× Z ∞
(ar+b)/ √ 2cr
z n exp
− z 2 2
dz, (5)
where we have substituted z = √
2cr{y + (ar + b)/(2cr)}. The result in (4) follows
immediately from (5).
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 2.2. If X and Y are jointly distributed according to (1) then f P (p) = 2K exp(−cp)K 0
2 p
abp
(6) for 0 < p < ∞.
Proof. From (1), the joint pdf of (X, P ) = (X, XY ) becomes f (x, 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) Z ∞
0
x −1 exp
−ax − bp x
dx. (7)
The result of the theorem follows by using Lemma 1.2 to calculate the above inte-
gral.
0.0 0.2 0.4 0.6 0.8 1.0
0.51.52.5
(a)
w
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(b)
w
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1234
(c)
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1234
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1234
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(d)
w
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1234
0.0 0.2 0.4 0.6 0.8 1.0
1234
0.0 0.2 0.4 0.6 0.8 1.0
1234
Figure 1. Plots of the pdf of (2) for (a): c = 0.1; (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).
0.0 1.0 2.0 3.0
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p
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(b)
p
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(c)
p
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(d)
p
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