Statistics of Correlated Generalized Gamma Variates

Transcription

1 Statistics of Correlated Generalized Gamma Variates Petros S. Bithas 1 and Theodoros A. Tsiftsis 2 1 Department of Electrical & Computer Engineering, University of Patras, Rion, GR-26500, Patras, Greece, ( pbithas@space.noa.gr) 2 Division of Telecommunications, Electrical and Computer Engineering Department, Aristotle University of Thessaloniki, Thessaloniki, Greece ( thtsif@auth.gr) Abstract. In this paper the most important statistical metrics of the product and the ratio of two correlated Generalized Gamma (GG) random variables (RV)s, raised to an arbitrary power, are studied. The probability density function (PDF) and the cumulative distribution function (CDF) have been derived in most cases in closed form. Moreover, an upper bound for the CDF of the sum of the two correlated GG RVs has been also obtained. Finally several numerical evaluated results have been also obtained. Keywords: Correlated statistics, Generalized Gamma. Extended Abstract The Generalized Gamma (GG) distribution was first introduced by Stacy [Stacy1962] back in 1962 as a generalization of the Gamma distribution. This versatile and generic distribution is especially attractive because it contains several distribution as special cases, e.g., Rayleigh, Nakagami-m [Nakagami1960] and Weibull [Weibull1951] and the well known lognormal as a limiting case. The GG distribution has many applications including life data, speech recognition, ultrasonic backscatter signals. More recently the GG has been applied to the context of wireless communications for accurately modeling short term fading in conjunction with long term (shadowing) fading channels [Coulson et al.1998]. Representative past works concerning distributions of products and ratios can be found in [Mathai1972], [Malik and Trudel1986], [Lee et al.1979], [Simon2002]. Despite the fact that the bivariate correlated Rayleigh, Nakagami, Weibull and Lognormal distribution have been studied many times [Tan and Beaulieu1997], [Sagias and Karagiannidis2005], [Alouini and Simon2002], the bivariate GG distribution has been introduced very recently [Piboongungon et al.2005] and has been applied to correlated fading channels with arbitrary or exponential correlation [Aalo and Piboongungon2005]. However to the

2 2 Bithas and Tsiftsis best of our knowledge the subject of distributions of products and ratios of GG random variables (RV)s has not been addressed in the open technical literature, and this is the subject of our paper. In this paper capitalizing on the bivariate GG distribution presented in [Piboongungon et al.2005], the most important statistical metrics, namely the probability density function (PDF) and cumulative distribution function (CDF), of the distribution of the products and the ratios of two correlated GG RVs have been obtained. Moreover, using an inequality between the arithmetic and geometric mean a useful union bound for the distribution of the sum of two correlated GG RVs is also presented. Due to space limitations imposed by the submission requirements, we are only going to present the PDF and the CDF of the product of two correlated GG RVs and two representative examples for the PDF and CDF of the product. Let X l 0 (l = 1 and 2) be two correlated GG distributed RVs, not necessarily identically distributed, with the joint PDF given by [Piboongungon et al.2005, eq. (5)] f X1,X 2 (x 1, x 2 ) = β2 m m+1 (z 1 z 2 ) β(m+1)/2 1 ρ (1 m)/2 (Ω 1 Ω 2 ) (m+1)/2 (1 ρ)γ (m) [ ( )] exp m x β 1 + xβ 2 I m 1 2m ρ (x 1 x 2 ) β with β > 0 and m 1/2 being the shaping parameters, Ω l = E X β l > 0 being the scaling parameters (with E denoting expectation) and Γ ( ) being the Gamma function [Gradshteyn and Ryzhik2000, 8.310/1]. Moreover, 0 ρ < 1 is the correlation coefficient between X 2 1 and X 2 2 and I n ( ) the nth (n N) order modified Bessel of the first kind [Gradshteyn and Ryzhik2000, eq. (8.406)]. (1) Distribution of the Product of GG Rvs We define a RV G = (X 1 X 2 ) c (2) with c > 0 being an arbitrary constant value. Probability Density Function By using (1), (2) and [Papoulis2001, eq. (6-74)], making a change of variables and using [Gradshteyn and Ryzhik2000, 3.471/9] and after some straight

3 Statistics of Correlated Generalized Gamma Variates 3 forward mathematical manipulations, the PDF of G, f G (x), is given by 2 β m m+1 x β(m+1)/(2c) 1 f G (x) = c (1 ρ) ρ (m 1)/2 Γ (m) (Ω 1 Ω 2 ) (m+1)/2 [ ] 2m ρ I m 1 x β/(2c) K 0 [ 2m 1 ρ y β/(2c) ] (3) Ω1 Ω 2 with K n ( ) being the nth order modified Bessel of the second kind [Gradshteyn and Ryzhik2000, eq. (8.407)]. Cumulative Distribution Function Integrating (3) with respect to G, and after making a change of variables integrals of the following form need to be solved I = A 0 x m I m 1 (B x)k 0 (C x)dx. This type of integral is very difficult, if not impossible, to be solved in closed form. An mathematical more tractable solution is to employ the infinite series representation for the modified Bessel function of the first kind, [Gradshteyn and Ryzhik2000, eq. (8.445)]. Following such an approach and using [Wolfram2006, eq. ( )] and after some mathematical manipulations the CDF of G, F G (x), can be obtained as 2 ρ k m 2(k+m) x (k+m)β/c F G (x) = k=0 (1 ρ) 2k+m Γ (m)γ (m + k)k!(k + m) 2 (Ω 1 Ω 2 ) k+m [ m {(k 2 x β/c ] [ 2mx β/(2c) ] + m) 1 F 2 1; k + m + 1, k + m; (1 ρ) 2 K 0 Ω 1 Ω 2 (1 ρ) Ω 1 Ω 2 + m [ xβ/(2c) (1 ρ) 2mx β/(2c) ] K 1 Ω 1 Ω 2 (1 ρ) Ω 1 Ω 2 m 1 F 2 [1; 2 x β/c ]} k + m + 1, k + m + 1; (1 ρ) 2 Ω 1 Ω 2 (4) with p F q ( ) being the generalized hypergeometric function [Gradshteyn and Ryzhik2000, eq. (9.14/1)], where p, q are integers. In Figs. 1 and 2 the PDF and CDF of G, respectively, are plotted as function of x for c = 2, ρ = 0.5, Ω 1 = 1.1, and Ω 2 = 0.9. In Fig. 1 it is depicted that as β, m have low values, e.g., βm < 2, f G (x) tends to infinite as x approaches to zero, while f G (x) decreases monotonically with an increase of x. For βm > 2, f G (x) increases with an increase of x until it reaches a maximum and then decreases as x increases. In Fig. 2 the impact of β and m in the CDF of G is illustrated.

4 4 Bithas and Tsiftsis Fig. 1. PDF of the product of two correlated GG RVs for β = 1.5, 3 and m = 2, 3. Fig. 2. CDF of the product of two correlated GG RVs for β = 1.5, 3 and m = 2, 3. References [Aalo and Piboongungon2005]V. A. Aalo and T. Piboongungon. On the multivariate generalized gamma distribution with exponential correlation. In IEEE GLOBECOM, volume 3, November [Alouini and Simon2002]M.-S. Alouini and M. K. Simon. Dual diversity over correlated log-normal fading channels. 50(12): , December [Coulson et al.1998]a. J. Coulson, A. G. Williamson, and R. G. Vaughan. Improved fading distribution for mobile radio. IEE Proceedings, 145(3): , June [Gradshteyn and Ryzhik2000]I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series, and Products. Academic, New York, 6 edition, [Lee et al.1979]r. Y. Lee, B. S. Holland, and J. A. Flueck. Distribution of a ratio of correlated Gamma random variables. SIAM Journal on Applied Mathematics, 36(2): , [Malik and Trudel1986]H. J. Malik and R. Trudel. Probability density function of the product and quotient of two correlated exponential random variables. Canadian Mathematical Bulletin, 29: , [Mathai1972]A. M. Mathai. Products and ratios of generalized Gamma variates. Skandinavisk Aktuarietidskrift, 55: , 1972.

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