On the Ratio of Inverted Gamma Variates

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1 AUSTRIAN JOURNAL OF STATISTICS Volume ), Number 2, On the Ratio of Inerted Gamma Variates M. Masoom Ali 1, Manisha Pal 2, and Jungsoo Woo 3 Department of Mathematical Sciences, Ball State Uniersity, USA Department of Statistics, Uniersity of Calcutta, India Department of Statistics, Yeungnam Uniersity Gyongsan, South Korea Abstract: In this paper the distribution and moments of the ratio of independent inerted gamma ariates hae been considered. Unbiased estimators of the parameter inoled in the distribution hae been proposed. As a particular case, the ratio of independent Ley ariates hae been studied. Zusammenfassung: In diesem Aufsatz werden die Verteilung und Momente om Quotient unabhängiger inertierter Gammaariablen betrachtet. Unerzerrte Schätzer für den Parameter in der Verteilung werden orgeschlagen. Als ein spezieller Fall wird der Quotient unabhängiger Ley-Variablen untersucht. Keywords: Moments, Unbiased Estimator, Maximum Likelihood Estimator, Ley Variables. 1 Introduction The distribution of the ratio of random ariables 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 ariables include Mendelian inheritance ratios in genetics, mass to energy ratios in nuclear physics, target to control precipitation in meteorology, and inentory ratios in economics. The distribution of ratio of random ariables hae been studied by seeral 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, Proost 1989) for gamma family, Pham-Gia 2000) for beta family, among others. The distribution of the ratio of independent gamma ariates 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 ariates. In this paper we derie the distribution of the ratio V = X/X + Y ), where X and Y are independent inerted gamma ariates, each with two parameters. An inerted gamma distribution IGp, σ) is gien by fx; p, σ) = σp Γp) x p 1 e σ/x, x > 0, σ > 0, p > 0, where p is the shape parameter and σ the scale parameter. The moments of the distribution of the ratio hae been obtained. As a particular case, the ratio of independent Ley ariables has been considered. The Ley distribution is one of the few distributions that are stable and that hae probability density functions that are analytically expressible. Moments of the Ley distribution do not exist. But the distribution is found to be ery useful in analysis of stock prices and also in Physics for the study of dielectric susceptibility see Jurlewicz and Weron, 1993).

2 154 Austrian Journal of Statistics, Vol ), No. 2, Distribution of the Ratio of Inerted Gamma Variables Let X and Y be independent random ariables distributed as IGp, σ x ) and IGq, σ y ), respectiely. Then U = 1/X and W = 1/Y are independently distributed as Gammap, σ x ) and Gammaq, σ y ), respectiely. We note that V = X/X + Y ) = W/U + W ). Let T = U + W. Then V and T are jointly distributed with pdf f V,T, t) = σp xσ q y Γp)Γq) e t{σx+σy σx)} t p+q 1 q 1 1 ) p 1, 0 < < 1, t > 0. Hence the marginal pdf of V is gien by f V ) = q 1 1 ) p ρ ) p q ρ q Bq, p) ρ, 0 < < 1, ρ = σ x > 0. σ y 1) After some algebraic manipulation, and using formula in Gradshteyn and Ryhzik 1965), the cumulatie distribution function cdf) of V is obtained as F V ) = where 2F 1 a, b; c; x) = {1 + ρ1 )/} q 2F 1 q, 1 p; q + 1; {1 + ρ1 )/} 1 ), 0 < < 1, qbq, p) i=0 a) i b) i x i, a) i = a a + 1)... a + i 1), a) 0 = 1, c) i i! is the Gauss hypergeometric series. Using formula ) in Gradshteyn and Ryhzik 1965), formulas and in Abramowitz and Stegtun 1970), and the density 1), we obtain the moments of the ratio V = X/X + Y ) as Bq + k, p) 2F 1 k, p; p + q + k; ρ 1)/ρ), if ρ > 1, EV k Bq, p) ) = k Bq + k, p) 2) ρ 2F 1 k, q + k; p + q + k; 1 ρ), if 0 < ρ < 1. Bq, p) In order to estimate ρ, we make use of the following lemma. Lemma 2.1: Let R = V/1 V ). Then a) R is distributed as the ratio of two independent random ariables with distributions Gammaq, σ y ) and Gammap, σ x ). b) ER k ) = ρ k Bq + k, p k)/bq, p), proided p > k. Proof: We hae R = V/1 V ) = X/Y = Y 1 /X 1. Since X and Y are independently distributed as IGp, σ x ) and IGq, σ y ), respectiely, a) easily follows. The distribution of R is therefore defined by the pdf f R r) = 1 r q 1 ρ q Bq, p) 1 + r/ρ), r > 0. p+q

3 M. M. Ali et al. 155 Hence, the k-th moment of R comes out to be ER k ) = Bq + k, p k) ρ k, proided p > k. Bq, p) From the lemma, for p > 1, we hae ) V ER) = E 1 V = q p 1 ρ 3) so that ER) 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 gien by ˆρ = p 1 nq n i=1 V i 1 V i, if p > 1. The ariance of this estimator is arˆρ) = p + q 1 nqp 2) ρ2, for p > 2. On the basis of independent random samples X 1,..., X n1 and Y 1,..., Y n2 drawn from the distributions of X and Y, respectiely, the maximum likelihood estimator MLE) of ρ is ρ = σ x / σ y, where σ x and σ y are the MLEs of σ x and σ y gien by σ x = n 1 p n 1, σ y = 1/X i ) i=1 n 2 q n 2. 1/Y i ) i=1 Noting that U = n 1 i=1 1/X i) and W = n 2 i=1 1/Y i) are independently distributed as Gamman 1 p, σ x ) and Gamman 2 q, σ y ), respectiely, Z = 1/U)/1/U + 1/W ) is distributed with pdf gien by 1) where p and q are replaced by n 1 p and n 2 q, respectiely. Also, ρ = n 1p Z n 2 q 1 Z, such that, from 3), Hence, is an unbiased estimator of ρ with E ρ) = n 1p n 1 p 1 ρ. ρ = n 1p 1 ρ 4) n 1 p ar ρ) = n 1p + n 2 q 1 n 2 qn 1 p 2) ρ2. 5) It can be easily seen that for n 1 = n 2 = n we get arˆρ) > ar ρ).

4 156 Austrian Journal of Statistics, Vol ), No. 2, Figure 1: Plots of the pdf 6) for ρ = 1, 2, 5. 3 Distribution of the Ratio of Ley Variables For p = q = 1/2, X and Y are two independent Ley ariables with scale parameters σ x and σ y, respectiely. The pdf and cdf of V then reduces to f V ) = 1 π ρ 1/2 1 ) 1/ ρ ) 1 ρ, 6) ρ = π 3/2 1 ) 1/2 1 + ρ 1 ) 1, 0 < < 1, ρ = σ x > 0 σ y and F V ) = 2 π = 2 ρ π 1 + ρ 1 1 ) 1/2 = 2 1 π sin ρ ρ 1 2F 1 1/2, 1/2; 3/2; 1 + ρ1 )/) 1 ) ) 1 2F 1 1, 1; 3/2; 1 + ρ1 )/) 1 ) 7), 0 < < 1. 8) The expression 7) of the cdf is obtained using formulas ) and ) in Gradshteyn and Ryhzik 1965), and expression 8) follows from formula in Abramowitz and Stegtun 1970). From 2) the moments of the distribution are Bk + 1/2, 1/2) 2F 1 k, 1/2; 1 + k; ρ 1)/ρ) if ρ > 1 EV k ) = π ρ k Bk + 1/2, 1/2) 2F 1 k, k + 1/2; 1 + k; 1 ρ) if 0 < ρ < 1. π

5 M. M. Ali et al. 157 From Abramowitz and Stegtun 1970), we hae the following useful relations for the hypergeometric function 2 F 1 a, b; c; z): Lemma 3.1: i) Formula ) ii) Formula ) n z n 2 F 1 a, b; c; z) = a) nb) n c) n 2F 1 a + n, b + n; c + n; z) 2F 1 a, a + 1/2; 1 + 2a; z) = 2 2a z) 2a iii) Formula ) 2F 1 a, a + 1/2; 2a; z) = 2 2a 1 1 z1 + 1 z) 2a 1 i) Formula ) 2F 1 a, b; c; z) = 1 z) b 2F 1 b, c a; c; z/z 1)). From Lemma 3.1 we get the following lemmas: Lemma 3.2: a) b) 2F 1 1, 3/2; 2; z) = 2 1 z1 + 1 z)) 2F 1 2, 5/2, 3; z) = z 3 1 z) 3/ z). 2 Proof: a) follows from Lemma 3.1iii) by substituting a = 1. By Lemma 3.1i) we get 2F 1 2, 5/2; 3; z) = 4 3 z 2 F 1 1, 3/2; 2; z). Hence, using a), we hae b). Lemma 3.3: a) 2F 1 1/2, 1; 2; ρ 1)/ρ) = 2 ρ 1 + ρ

6 158 Austrian Journal of Statistics, Vol ), No. 2, b) 2F 1 1/2, 2; 3; ρ 1)/ρ) = ρ 2 2F 1 2, 5/2; 3; 1 ρ) = 4 3 ρ1 + 2 ρ) 1 + ρ) 2. Proof: a) follows from Lemma 3.1ii) by substituting a = 1/2, z = ρ 1)/ρ, and b) follows from Lemma 3.1i) and Lemma 3.2b) by taking a = 1/2, b = 2, c = 3, and z = ρ 1)/ρ. Using Lemmas 3.2 and 3.3 we hae such that EV ) = ρ 1 + ρ, EV 2 ) = arv ) = ρ1 + 2 ρ) 21 + ρ) 2, 9) ρ 21 + ρ) 2. 10) If ρ denotes the MLE of ρ based on independent random samples X 1,..., X n1 and Y 1,..., Y n2 from the distributions of X and Y, then, from 4) and 5), an unbiased estimator of ρ and its ariance are gien by n 1 ρ = n 1 2 ρ n 1 ) 2 n1 2 2n 1 + n 2 2) ar ρ) = ρ 2, for n 1 > 4. n 2 n 1 4) Since ρ = ρ/1 + ρ) is a monotone increasing and bounded function of ρ, inference on ρ will be equialent to inference on ρ. Hence, from any estimator of ρ we can obtain an estimator of ρ by a one-to-one transformation. From 9) and 10), for a random sample V 1,..., V n of size n from the distribution 6), an unbiased estimator of ρ is V = n 1 n i=1 V i with ariance ρ 1 ρ )/2n. The corresponding estimator of ρ is ˆρ = V /1 V )) 2 with asymptotic ariance 2ρ 3/2 1 + ρ) 2 /n. Acknowledgement The authors thank the referee for his/her suggestions, which immensely helped to improe the presentation of the paper. References Abramowitz, M., and Stegtun, I. A. 1970). Handbook of Mathematical Functions. New York: Doer Publication Inc. Ali, M. M., Woo, J., and Pal, M. 2006). Distribution of the ratio of generalized uniform ariates. Pakistan Journal of Statistics, 22, Basu, A. P., and Lochner, R. H. 1979). On the distribution of the generalized life distributions. Technometrics, 33,

7 M. M. Ali et al. 159 Bowman, K. O., Shenton, L. R., and Gailey, P. C. 1998). Distribution of the ratio of gamma ariates. Communications in Statistics Simulation and Computation, 27, Gradshteyn, I. S., and Ryhzik, I. M. 1965). Tables of Integrals, Series and Products. New York: Academic Press. Jurlewicz, A., and Weron, K. A. 1993). Relationship between asymmetric Ley-stable distributions and the dielectric susceptibility. Journal of Statistical Physics, 73, Korhonen, P. J., and Narula, S. C. 1989). The probability distribution of the ratio of the absolute alues of two normal ariables. Journal of Statistical Computation and Simulation, 33, Marsaglia, G. 1965). Ratios of normal ariables and ratios of sums of uniform ariables. Journal of the American Statistical Association, 60, Pham-Gia, T. 2000). Distributions of the ratios of independent beta ariables and applications. Communications in Statistics Theory and Methods, 29, Press, S. J. 1969). The t ratio distribution. Journal of the American Statistical Association, 64, Proost, S. B. 1989). On the distribution of the ratio of powers of sums of gamma random ariables. Pakistan Journal of Statistics, 5, Authors addresses: M. Masoom Ali Department of Mathematical Sciences Ball State Uniersity Muncie, IN USA mali@bsu.edu Manisha Pal Department of Statistics Uniersity of Calcutta 35 Ballygunge Circular Road Kolkata India manishapal2@gmail.com Jungsoo Woo Department of Statistics Yeungnam Uniersity Gyongsan South Korea

On the Ratio of two Independent Exponentiated Pareto Variables

AUSTRIAN JOURNAL OF STATISTICS Volume 39 21), Number 4, 329 34 On the Ratio of two Independent Exponentiated Pareto Variables M. Masoom Ali 1, Manisha Pal 2 and Jungsoo Woo 3 1 Dept. of Mathematical Sciences,