On the Ratio of Inverted Gamma Variates

The distribution of the ratio of random variables are of interest in problems in biological and physical sciences, econometrics, classification, and ranking and selection. 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 distribution of ratio of random variables have been studied by several authors like Marsaglia (1965) and Korhonen and Narula (1989) for normal family, Press (1969) for student’s t family, Basu and Lochner (1979) for Weibull family, Provost (1989) for gamma family, Pham-Gia (2000) for beta family, among others. The distribution of the ratio of independent gamma variates with shape parameters equal to 1 was studied by Bowman, Shenton, and Gailey (1998). Recently, Ali, Woo, and Pal (2006) obtained the distribution of the ratio of generalized uniform variates. In this paper we derive the distribution of the ratio V = X/(X + Y ), where X and Y are independent inverted gamma variates, each with two parameters. An inverted gamma distribution IG(p, σ) is given by


Introduction
The distribution of the ratio of random variables are of interest in problems in biological and physical sciences, econometrics, classification, and ranking and selection.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 distribution of ratio of random variables have been studied by several authors like Marsaglia (1965) and Korhonen and Narula (1989) for normal family, Press (1969) for student's t family, Basu and Lochner (1979) for Weibull family, Provost (1989) for gamma family, Pham-Gia (2000) for beta family, among others.The distribution of the ratio of independent gamma variates with shape parameters equal to 1 was studied by Bowman, Shenton, and Gailey (1998).Recently, Ali, Woo, and Pal (2006) obtained the distribution of the ratio of generalized uniform variates.
In this paper we derive the distribution of the ratio V = X/(X + Y ), where X and Y are independent inverted gamma variates, each with two parameters.An inverted gamma distribution IG(p, σ) is given by where p is the shape parameter and σ the scale parameter.
The moments of the distribution of the ratio have been obtained.As a particular case, the ratio of independent Levy variables has been considered.The Levy distribution is one of the few distributions that are stable and that have probability density functions that are analytically expressible.Moments of the Levy distribution do not exist.But the distribution is found to be very useful in analysis of stock prices and also in Physics for the study of dielectric susceptibility (see Jurlewicz and Weron, 1993).

Distribution of the Ratio of Inverted Gamma Variables
Let X and Y be independent random variables distributed as IG(p, σ x ) and IG(q, σ y ), respectively.Then U = 1/X and W = 1/Y are independently distributed as Gamma(p, σ x ) and Gamma(q, σ y ), respectively.We note that Hence the marginal pdf of V is given by After some algebraic manipulation, and using formula 8.391 in Gradshteyn and Ryhzik (1965), the cumulative distribution function (cdf) of V is obtained as is the Gauss hypergeometric series.Using formula 3.197(3) in Gradshteyn and Ryhzik (1965), formulas 15.3.3 and 15.3.5 in Abramowitz and Stegtun (1970), and the density (1), we obtain the moments of the ratio V = X/(X + Y ) as (2) In order to estimate ρ, we make use of the following lemma.
Lemma 2.1: R is distributed as the ratio of two independent random variables with distributions Gamma(q, σ y ) and Gamma(p, σ x ).
Since X and Y are independently distributed as IG(p, σ x ) and IG(q, σ y ), respectively, (a) easily follows.
The distribution of R is therefore defined by the pdf Hence, the k-th moment of R comes out to be From the lemma, for p > 1, we have so that E(R) • (p − 1)/q = ρ.Thus, for a random sample V 1 , . . ., V n of size n from the distribution of V , an unbiased estimator of ρ will be given by The variance of this estimator is On the basis of independent random samples X 1 , . . ., X n 1 and Y 1 , . . ., Y n 2 drawn from the distributions of X and Y , respectively, the maximum likelihood estimator (MLE) of ρ is ρ = σx /σ y , where σx and σy are the MLEs of σ x and σ y given by .
(1/Y i ) are independently distributed as Gamma(n 1 p, σ x ) and Gamma(n 2 q, σ y ), respectively, Z = (1/U )/(1/U + 1/W ) is distributed with pdf given by (1) where p and q are replaced by n 1 p and n 2 q, respectively.Also, is an unbiased estimator of ρ with It can be easily seen that for n 1 = n 2 = n we get var(ρ) > var( ρ).