On the Distribution of the Product and Ratio of Independent Generalized Gamma-Ratio Random Variables

Transcription

1 Sankhyā : The Indian Journal of Statistics 2007, Volume 69, Part 2, pp c 2007, Indian Statistical Institute On the Distribution of the Product Ratio of Independent Generalized Gamma-Ratio Rom Variables Carlos A. Coelho João T. Mexia The New University of Lisbon, Portugal Abstract Using a decomposition of the characteristic function of the logarithm of the product of independent generalized gamma-ratio rom variables (r.v. s), we obtain explicit expressions for both the probability density cumulative distribution functions of the product of independent r.v. s with generalized F or generalized gamma-ratio distributions in the form of particular mixtures of generalized Pareto inverted Pareto distributions. The expressions obtained do not involve any unsolved integrals are convenient for computer implementation. By considering power parameters which are not required to be positive, we were able to obtain, as particular cases, not only the distributions for the product of folded T folded Cauchy r.v. s but also for the ratio of two independent products of generalized gamma-ratio r.v. s. Theoretical applications of the results as well as simulations are presented. AMS (2000) subject classification. Primary 62E5; secondary 62E0. Keywords phrases. Particular mixtures, Pareto inverted Pareto distributions, GIG distribution, sum of exponentials, difference of exponentials, folded T, folded Cauchy, beta prime, beta second kind. Introduction The problem of obtaining an explicit expression, without involving any unsolved integrals, for both the probability density function (p.d.f.) the cumulative distribution function (c.d.f.) of the product of independent generalized gamma-ratio rom variables (r.v. s) or generalized F r.v. s, as they are also called, is a challenging one, especially since the characteristic function is not readily available for such r.v. s. In this paper, we determine the distribution of the product of independent generalized gamma-ratio (GGR)

2 222 Carlos A. Coelho João T. Mexia rom variables under the form of particular mixtures of Pareto inverted Pareto distributions. Expressions for the p.d.f. of such a product were obtained by Shah Rathie (974) in terms of Fox s H function. Also Pham-Gia Turkkan (2002) obtained expressions for the p.d.f. of the product of only two independent generalized F r.v. s in terms of the LauricellahypergeometricD-function of two variables. However, even nowadays, when good softwares for symbolic numeric computation are available, Fox s H function the Lauricella function are not readily computable, being usually computed in terms of the integrals that define them. Although Pham-Gia Turkkan (2002) developed an efficient computer code to compute the Lauricella hypergeometric function, their approach is not extensible to the product of more than two r.v. s these authors point out that when we consider the product of more than two GGR r.v. s, it seems that frequently, however, no closed form solution for these operations can be obtained, one has to resort to approximate approaches, including simulation. The same authors, when referring to the results in Shah Rathie (974) say that these results, although very convenient notationwise, are difficult to be programmed on a computer hence are difficult to be used in applications. Yet the same authors say that they are, however, essential when the number of variables in the product, or quotient, is larger than 2. In this paper, our aim is to obtain explicit simpler expressions for the p.d.f. the c.d.f. of the product of independent generalized gammaratio r.v. s which may be readily implemented computationally that, given its structure, may also give us ready access to asymptotic nearexact distributions. Given the approach followed, not only the distributions for the non-central case are readily at h but in addition the distribution for the product of any non-null power of GGR r.v. s is also available. As particular immediate cases, we have the product of independent generalized second kind beta or beta prime, folded T folded Cauchy r.v. s yet, of course, F r.v. s. Given the fact that the characteristic functions for the GGR rom variables are not readily available given the fact that we are dealing with a product of rom variables, it is more convenient to carry out our work through the decomposition of the characteristic function of the logarithm of the product of the GGR rom variables. Another novelty is that, although usually only positive power parameters are considered for the GGR distributions, nothing actually forces those parameters to be positive. In fact, to consider a negative power parameter

3 Product ratio of gamma-ratio rom variables 223 in the GGR distribution is clearly equivalent to considering the reciprocal of that given rom variable with the symmetrical positive power parameter. Given the way the problem is approached, even negative power parameters may be easily considered in the GGR distributions also the distribution of the ratio of two GGR rom variables or of the ratio of two products of GGR rom variables are particular cases of the results obtained in the paper. Products of several independent GGR rom variables arise in several areas of application in statistics as it is shown in Section 4 are also related to a test statistic used in the multivariate linear functional model (Provost, 986). 2 Some Preliminary Results 2.. The generalized gamma-ratio (GGR) distribution. In order to establish some of the notation, nomenclature a result used, we will start by defining what we mean by a generalized gamma-ratio (GGR) distribution. Let X Γ(r,λ ) X 2 Γ(r 2,λ 2 ) be two independent r.v. s with gamma distributions with shape parameters r r 2 rate parameters λ λ 2, that is, for example, X has p.d.f. Let then f X (x) = λr Γ(r ) exp( λ x) x r, r,λ > 0; x>0. Y = X /β, Y 2 = X /β 2, β IR\{0} Z = Y /Y 2. We will say that Y Y 2 have generalized gamma distributions that Z has a GGR distribution. Using stard methods, we have the p.d.f. s of Y i (i =, 2) Z given by f Yi (y i )= β λr i ) i ( λ Γ(r i ) exp i y β i y βr i i, y i > 0, (i =, 2) f Z (z) = β kr B(r,r 2 ) ( +kz β) r r 2 z βr, z > 0,

4 224 Carlos A. Coelho João T. Mexia where k = λ /λ 2,B(, ) is the beta function. We will denote the fact that Z has the GGR distribution with parameters k, r, r 2 β by Z GGR(k, r,r 2,β). The hth moment of Z is easily derived as ( E Z h) = k h/β Γ(r + h/β) Γ(r 2 h/β), Γ(r ) Γ(r 2 ) ( r <h/β<r 2 ). (2.) If β =,k = m/n, r = m/2 r 2 = n/2, with m, n IN, then Z has an F distribution with m n degrees of freedom. This is the reason why the distribution of Z is also called a generalized F distribution (Shah Rathie, 974) The generalized integer gamma (GIG) distribution. In this subsection in the two following ones, we will establish some distributions that will be used in the next section. Let X j Γ(r j,λ j ), j =,...,p, be p independent r.v. s with gamma distributions with shape parameters r j IN rate parameters λ j > 0(j =,...,p). We will say that then the r.v. Y = has a GIG distribution of depth p, with shape parameters r j rate parameters λ j,(j =,...,p), we will denote this fact by X j Y GIG(r j,λ j ; p). The p.d.f. the c.d.f. of Y are respectively given by (Coelho, 998) f Y (y) =K P j (y) exp( λ j y) (2.2) F Y (y) = K Pj (y) exp( λ j y), (2.3)

5 Product ratio of gamma-ratio rom variables 225 where with c j,rj = K = p P j (y) = (r j )! λ r j j, P j (y) = r j k= r j k= k c j,k (k )! i=0 c j,k y k (2.4) y i i! λ k i j p (λ i λ j ) r i, j =,...,p, (2.5) i= i j, c j,rj k = k where k i= (r j k + i )! (r j k )! R(i, j, p) = R(i, j, p) c j,rj (k i), (k =,...,r j ) (j =,...,p), (2.6) r k (λ j λ k ) i (i =,...,r j ). (2.7) k= k j 2.3. The distribution of the sum of rom variables with exponential distribution the distribution of the difference of two of these rom variables. Let X i Exp(λ i ) Γ(,λ i ), i =,...,p, be p independent exponential r.v. s with rate parameters λ i (i =,...,p), let Y = X i. i= The distribution of the r.v. Y is a particular case of the GIG distribution (Coelho, 998) of depth p, with all the shape parameters equal to, whose p.d.f. may be written as f Y (y) =K c j, exp( λ j y), (2.8)

6 226 Carlos A. Coelho João T. Mexia where K = p λ j c j, = p k= k j We will denote the fact that Y has this distribution by Theorem 2.. Let be two independent r.v. s let λ k λ j (j =,...,p). (2.9) Y SE(λ j ; p) GIG(,λ j ; p). (2.0) Y SE(λ j ; p) Y 2 SE(ν j ; p) (2.) Z = Y Y 2. The p.d.f. c.d.f. of Z are given by K K 2 H 2j d j, exp(ν j z), z 0 f Z (z) = K K 2 H j c j, exp( λ j z), z 0 F Z (z) = K K 2 K K 2 respectively, where, for j =,...,p, H j = c j, = h= (2.2) H 2j d j, ν j exp(ν j z), z 0 ( ) d j, c j, H 2j + H j ( exp( λ j z)), z 0 ν j λ j (2.3) K = p k= k j p λ j, K 2 = λ k λ j, d j, = p ν j (2.4) p k= k j d h, λ j + ν h H 2j = ν k ν j, (2.5) h= c h, λ h + ν j. (2.6)

7 Product ratio of gamma-ratio rom variables 227 Proof. The p.d.f. s of Y Y 2 may then be written as f Y (y )=K c j, exp( λ j y ) f Y2 (y 2 )=K 2 d j, exp( ν j y 2 ) (2.7) respectively, with K K 2 given by (2.4), c j, d j, given by (2.5) H j H 2j given by (2.6), so that the p.d.f. of Z will be given by + f Z (z)= K K 2 c j, exp( λ j y ) d j, exp{ ν j (y z)} dy where max(z,0) = K K 2 k= + exp(ν k z) c j, d k, exp{ (λ j + ν k )y } dy = max(z,0) exp{ (λ j + ν k )y } dy, λ j +ν k, z 0, max(z,0) exp{ (λ j +ν k )z} λ j +ν k, z 0 so that, after some small rearrangements, we have f Z (z) given by (2.2). The c.d.f. of Z is then easily derived from (2.2) as given by (2.3). Corollary 2.. If we take, for some k>0, W = k exp(z), then, the p.d.f. the c.d.f. of W will be given by K K 2 H j c j, (kw) λ j w, w k, f W (w) = K K 2 H 2j d j, (kw) ν j w, 0 <w k, F W (w) = respectively. K K 2 K K 2 ( H 2j d j, ν j + H j c j, λ j ( (kw) λ j) ), w k, (2.8) H 2j d j, ν j (kw) ν j, 0 <w k, (2.9)

8 228 Carlos A. Coelho João T. Mexia The distribution of Z is also the distribution of the sum of p independent r.v. s with the distribution of the difference of two independent r.v. s with exponential distribution, either with similar or different parameters, i.e., the distribution of the sum of p independent r.v. s with Laplace or generalized Laplace distributions. We should note that although, namely in (2.4), it may seem that it would not be reasonable to take p, as a matter of fact in both (2.2) (2.8) taking p will yield genuine p.d.f. s (see Appendix). Taking into account that if the r.v. X has an exponential distribution with rate parameter λ p.d.f. f X (x) =λ exp( λx), x > 0; λ>0, for k > 0, the r.v. Y = k exp(x) has a Pareto distribution with rate parameter λ, lower bound parameter k, p.d.f. ( y ) λ f Y (y) =λ, y k; λ>0. k y We will say that the r.v. X = X with p.d.f. f X (x) =λ exp(λx), x < 0; λ>0 has a symmetric exponential distribution with rate parameter λ that the r.v. Y =/Y =(/k) exp(x )=(/k) exp( X), with p.d.f. ( y ) λ f Y (y) =λ k y, 0 <y k ; λ>0, has an inverted Pareto distribution with rate parameter λ lower bound parameter k. We may then note that the distribution of Z, forz 0, may be seen as a particular mixture of exponential distributions with rate parameters λ j (j =,...,p), with weights p j = K K 2 H j c j λ j, j =,...,p, such that p j = P [Z 0];

9 Product ratio of gamma-ratio rom variables 229 for z 0, as a particular mixture of symmetric exponential distributions with rate parameters ν j,withweights such that q j = K K 2 H 2j d j ν j, q j = P [Z 0]. j =,...,p, On the other h, the distribution of W may, for w k, be seen as a particular mixture of Pareto distributions with rate parameters λ j (j =,...,p) lower bound parameters k,withweightsp j (j =,...,p), such that p j = P [W k ], for w k, it may be seen as a mixture of inverted Pareto distributions with rate parameters ν j (j =,...,p) lower bound parameters k, with weights q j (j =,...,p), such that q j = P [W k ] The distribution of the difference of two GIG distributions. Theorem 2.2. Let, for j =,...,p l =,...,p 2, Y GIG(r j,λ j ; p ), Y 2 GIG(r 2l,ν l ; p 2 ), be two independent r.v. s, let Z = Y Y 2. Then, by taking K c jk defined in a manner similar to the definitions of K c jk in (2.4) (2.5)-(2.7) respectively, K 2 d lh defined in a corresponding manner, replacing p by p 2, r j by r 2l λ j by ν l, the p.d.f. of Z is given by f Z (z) = p K K 2 P j (z) exp( λ j z), z 0, p K K 2 P2j (z) exp(ν j z), z 0, (2.20)

10 230 Carlos A. Coelho João T. Mexia where P j (z) = r j k= c jk p 2 r 2l l= h= ( ) k k i (h + i )! z i (λ j + ν l ) h+i (2.2) d lh k i=0 P 2j (z) = r j k= c jk p 2 r 2l l= h= ( ) h h i (k + i )! ( z) ; (2.22) i (λ j + ν j ) k+i d lh h i=0 the c.d.f. by F Z (z) = p K K 2 Pj (z) exp( λ j z), z 0, p K K 2 P2j (z) exp(ν j z), z 0, (2.23) with P j (z) = r j k= c jk p 2 r 2l l= h= ( k i d lh k i=0 ) (h+i )! (λ j +ν l ) h+i k i t=0 (k i)! t! z t λ k i t j (2.24) P 2j (z) = r j k= c jk p 2 r 2l l= h= ( h i d lh h i=0 ) (k+i )! (λ j +ν j ) k+i h i t=0 (h i)! t! ( z) t ν h i t l (2.25). Proof. Considering (2.2) taking K c jk defined in a manner similar to the definitions of K c jk in (2.4) (2.5)-(2.7) respectively, K 2 d lh defined in a corresponding manner, the p.d.f. of Z is given

11 by Product ratio of gamma-ratio rom variables 23 f Z (z) = K K 2 + = K K 2 + p ( rij max(z,0) k= { p2 ( r2l p = K K 2 l= p p 2 max(z,0) l= p 2 l= + ) c jk y k exp( λ j y ) ) } d lh (y z) h exp { ν l (y z)} dy h= ( rj k= c jk y k )( r2l ) d lh (y z) h h= exp{ (λ j + ν l )y } exp(ν l z) dy r j r 2l max(z,0) k= h= c jk d lh y k (y z) h exp{ (λ j + ν l )y } exp(ν l z) dy p p 2 r j r 2l h ( h = K K 2 exp(ν l z) c jk d lh i l= where, for m>0k IN 0, + max(z,0) exp( my) y k dy = k= h= + max(z,0) exp( mz) i=0 ) ( z) h i exp{ (λ j +ν l )y } y k+i dy, (2.26) k i=0 k! i! z i, z 0, mk i+ k!, z 0, mk+ (2.27) (which is, for z>0, a particular version of an incomplete gamma function) yielding (2.20) with Pj (z) givenby Pj (z) r j p 2 r 2l h ( k+i h = c jk d lh )( z) h i (k+i )! z t i t! (λ k= l= h= i=0 t=0 j + ν l ) k+i t. (2.28) It is however interesting useful to observe that, given that the distribution of Y Y 2 Y 2 Y are symmetric, that we may in (2.26)

12 232 Carlos A. Coelho João T. Mexia integrate with respect to y 2 instead of y, we may obtain the p.d.f. of Z given by an expression similar to the one in (2.20) with Pj (z) givenby (2.2) P2j (z) r j p 2 r 2l k ( h+i k = c jk d lh )z k i (h + i )! z t i t! (λ k= l= h= i=0 t=0 j + ν l ) h+i t. (2.29) We may further note that h ( h i i=0 k ( k = i i=0 )( z) h i k+i t=0 ) z k i (k+i )! t! (h+i )! (λ j +ν l ) h+i. z t (λ j +ν l ) k+i t This way, in order to obtain a simpler expression for the p.d.f. of Z, wemay consider the p.d.f. in (2.20) with Pj (z) given by (2.2) P 2j (z) given by (2.22). Then, from (2.20) through (2.22), it is easy to derive the c.d.f. of Z as given by (2.23) through (2.25). Corollary 2.2. If we consider, for some k>0, the r.v. W = k exp(z), we have f W (w) = F W (w) = p K K 2 Pj (log(kw)) (kw) λ j w, w k, p K K 2 P2j (log(kw)) (kw) ν j w, 0 <w k, p K K 2 Pj (log(kw)) (kw) λ j, w k, p K K 2 P2j (log(kw)) (kw) ν j, 0 <w k. (2.30) (2.3)

13 Product ratio of gamma-ratio rom variables 233 As we did with (2.2) (2.8) also in (2.20) through (2.3), we may take both p p 2, which still yields genuine distributions (see Appendix). 3 The Distribution of the Product of m Independent Rom Variables with GGR Distributions 3.. The case with all distinct shape parameters for the numerator the denominator. In this section, we will obtain explicit concise expressions for the p.d.f. the c.d.f. of the r.v. W = m X j, where the X j s are independent r.v. s with all distinct shape parameters. The p.d.f. c.d.f. of W are stated in the following Theorem. Theorem 3.. Let X j GGR(k j,r j,r 2j,β j ), j =,...,m, be m independent r.v. s, where the power parameters β j s (j =,...,m) are not necessarily all positive. Let { { rj if β βj j > 0, r2j if β j > 0, = β j, s j = s 2j = r 2j if β j < 0, r j if β j < 0. (3.) Then, if βj s j βk s k βj s 2j βk s 2k, j k, j, k {,...,m}, the p.d.f. the c.d.f. of the r.v. are given by f W (w) = lim n K K 2 lim n K K 2 m h=0 m h=0 W = m X j, (3.2) n H 2jh d hj (w/k ) s jh w, 0 <w K, n H jh c hj (w/k ) s 2jh w, w K, (3.3)

14 234 Carlos A. Coelho João T. Mexia m lim K K 2 n F W (w) = m lim K K 2 n n d ( hj w ) s jh H 2jh K, 0 <w K, h=0 { n h=0 s jh d hj c hj H 2jh s + H jh jh s 2jh ( ( w K ) s 2jh ) }, w K (3.4) respectively, where K = m h=0 K = m n s jh, K 2 =, for j =,...,m n =0,,..., m n c hj = s 2ην, d hj = s 2jh η= ν=0 (η j ν h) k /β j j, (3.5) m h=0 m η= ν=0 (η j ν h) n s 2jh, (3.6) n s ην, (3.7) s jh with m n d kl m H jh = s 2jh +, H 2jh = s kl k= l=0 k= l=0 n c kl s jh +, (3.8) s 2kl s jh = β j (s j + h) s 2jh = β j (s 2j + h) (j =,...,m). (3.9) Proof. From (2.), we have so that if we take then we have ( ) E Xj h ( E Y h j = k h/β j j ) Γ(r j + h/β j )Γ(r 2j h/β j ), Γ(r j )Γ(r 2j ) Y j = k /β j j X j, = Γ(r j + h/β j )Γ(r 2j h/β j ) Γ(r j )Γ(r 2j ).

15 Product ratio of gamma-ratio rom variables 235 Thus, if we define then we have W = where we may take W = m m X j = m Y j, k /β j j } {{ } =K Z =logw = m Y j = K W, m log Y j, so that the characteristics function Φ Z (t) is given, under the form of a Mellin transform, for i =, by Φ Z (t) = = Using in (3.0), the relation m Φ log Yj (t) = m Γ(z) = exp( γz) z m E ( Yj it ) Γ(r j + it/β j )Γ(r 2j it/β j ) Γ(r j )Γ(r 2j ) i= ( ) i i + z exp(z/i),. (3.0) which is valid for any z /C\{0,, 2,...} (Ahlfors, 979, 2.4, pp. 98; Lang, 999, Chap. XV, 2, pp. 43), where γ is the Euler gamma constant, we may, after some simplifications, write m Φ Z (t) = β j (r j +k)(β j (r j +k)+it) β j (r 2j +k)(β j (r 2j +k) it). k=0 (3.) Since the β j s (j =,...,m) are not necessarily all positive, we take β j, s j s 2j as given by (3.), so that we may write Φ Z (t) = m k=0 βj (s j +k) ( βj (s j +k)+it ) β j (s 2j +k) ( βj (s 2j +k) it ), (3.2)

16 236 Carlos A. Coelho João T. Mexia where now β j (s j + k) > 0β j (s 2j + k) > 0 for all j =,..., k = 0,,... Expression (3.2) shows that, if β j s j β k s k β j s 2j β k s 2k, j k, j, k {,...,m}, the distribution of Z is the same as the distribution of a sum of infinitely many independent r.v. s distributed as the difference of two independent exponential distributions, with parameters β j (s 2j + k) β j (s j + k) (j =,...,m; k =0,,...). Alternatively, Z is distributed as the difference of two independent r.v. s, each one having the distribution of the sum of infinitely many independent exponential r.v. s. Thus, since W = K exp(z), (3.3) with K given by (3.5), taking s jh = β j (s j + k) s 2jh = β j (s 2j + k) using (2.8) (2.9) in Corollary 2. as a basis, we may write the p.d.f. the c.d.f. of W as in (3.3) through (3.9). To simplify the notation, we may replace the pair of indices (k, j) by h = km + j, setting s ih = β j (s ij + k), for i =, 2; j =,...,m; k =0,...,n. (3.4) We may now define K, K 2, c j, d j, H j H 2j (j =,...,m(n + )), (with n ) in a way similar to the one used in Theorem 3., that is,, for j =,..., K = m(n+) s j, K 2 = m(n+) s 2j, c j = m(n+) k= k j s j s k, d j = m(n+) k= k j s 2j s 2k, H j = m(n+) h= d h s j + s 2h, H 2j = m(n+) h= c h s h +, s 2j

17 Product ratio of gamma-ratio rom variables 237 so that we may, for example, write the c.d.f. of the r.v. W as m(n+) lim K K 2 n F W (w) = lim K K 2 n m(n+) d ( j w ) s j H 2j s j K, 0 <w K, [ d j H 2j s j { c j w ) } + H j ( ] s 2j s 2j K, w K. (3.5) 3.2. The general case. In the general case, we will allow for the possibility that some of the parameters s j = βj s j, j {i,...,m}, will be equal, that also some of the parameters s 2j = βj s 2j, j {i,...,m}, will be equal. More precisely, without any loss of generality, let us suppose that τ of the m parameters s j are equal to s, τ 2 are equal to s 2, so on, that τ p are equal to s p,withp <m p τ j = m, that, similarly, η of the m parameters s 2j are equal to s 2, η 2 are equal to s 22, so on, that η p2 are equal to s 2p2,withp 2 <m p 2 η j = m. Then the characteristic function in (3.0) may be written as Φ Z (t) = p k=0 ( sj + βj k ) τ j ( sj + βj k it ) τ j p 2 ( s2j + βj k ) η j ( s2j + βj k it ) η j = lim p n k=0 k=0 n ( sj + βj k ) τ j ( sj + βj k it ) τ j p 2 k=0 n ( s2j + βj k ) η j ( s2j + βj k it ) η j, that is, the characteristic function of the difference of two independent r.v. s with GIG distributions, where the first one, that is, the one with positive

18 238 Carlos A. Coelho João T. Mexia sign, with depth p (n + ) (with n ), with rate parameters associated shape parameters s + βk, }{{}..., s j + βj k }{{} k=0,...,n k=0,...,n τ,...,τ }{{} n+,..., τ j,...,τ j }{{} n+,..., s p + βp k }{{} k=0,...,n,..., τ p,...,τ p }{{} n+ the second one, that is, the one with negative sign, with depth p 2 (n+) (with n ), with shape parameters associated rate parameters s 2 + βk, }{{}..., s 2j + βj k }{{} k=0,...,n k=0,...,n η,...,η }{{},..., η j,...,η j }{{} n+ n+,..., s 2p2 + βp 2 k }{{} k=0,...,n,..., η p2,...,η p2 }{{} n+. Let us consider the vectors [ ] τ = τ,...,τ p,τ,...,τ p,...,τ,...,τ p, }{{} n+ times η = [ ] η,...,η p2,η,...,η p2,...,η,...,η p2, }{{} n+ times where, for j =,...,p (n +) l =,...,p 2 (n + ), with j = kp + h l = kp 2 + i, fork =0,...,n, h =,...,p i =,...,p 2, τ j = τ h, η l = η i, (similarly to the vectors s s 2 considered in the previous subsection) the vectors [ ] s = s,..., s p, s +β,..., s p +βp,..., s +βn,..., s p +βp n s 2 = [ ] s 2,..., s 2p, s 2 +β,..., s 2p +βp,..., s 2 +βn,..., s 2p +βp n where, once again, for j =,...,p (n +)l =,...,p 2 (n + ), with j, l, k, h i defined as above, s j = s h + β h k s 2l = s 2i + β i k.

19 Product ratio of gamma-ratio rom variables 239 The p.d.f. the c.d.f. of W = K exp(z), for K defined as in (3.3), may then be derived respectively from (2.30) (2.3) in Corollary 2.2, taking into account the shape the rate parameters mentioned above, as f W (w) = F W (w) = p 2 (n+) lim K K 2 n p (n+) lim K K 2 n p 2 (n+) lim K K 2 n p (n+) lim K K 2 n P 2j P j P 2j P j ( ( w )) ( w ) s 2j log K K w, 0 <w K ( ( w )) ( w ) s j log K K w, w K, (3.6) ( log ( w K )) ( w K ) s 2j, 0 <w K, ( log ( w K )) ( w K ) s j, w K, (3.7) where now K, K 2, Pj ( ) P2j ( ) are defined as in Theorem 2.2 Corollary 2.2, with p replaced by p (n +), p 2 replaced by p 2 (n +), λ j replaced by s j, ν l replaced by s 2j, r j replaced by τj r 2l replaced by ηl. 4 Applications 4.. Testing the (mutual) independence of a set of variables. Let us suppose that the rom vector X (p ) has a p-multivariate Normal distribution X N p (µ, Σ), we want to test the hypothesis that the p rom variables in X are mutually independent. This hypothesis may be written as Σ=diag(σ 2,...,σ 2 p). (4.) If we have a sample of size N (> p), the power 2/N of the likelihood ratio test statistic is (Anderson, 984, Chap. 9, sec. 9.2; Coelho, 992, 998, 2004) V Λ= p V, (4.2) jj

20 240 Carlos A. Coelho João T. Mexia where V is either the sample variance-covariance matrix or the maximum likelihood estimator of Σ, V jj is the j-th diagonal element of V.WithVj the corresponding matrix for the last p j variables (a lower right submatrix of V ), we have (see Anderson, 984, Chap. 9, Thm ; Coelho, 992, Chap. 4) p Λ= Λ j(j+,...,p) (4.3) with Vj Λ j(j+,...,p) = V jj Vj+ as the statistic for testing the independence between the j-th variable those with indices j +,...,p. Moreover, when H 0 holds, Λ j(j+,...,p) will have a beta distribution with parameters (n p + j)/2 (p j)/2, where n = N (Anderson, 984, Chap. 9, sec. 9.3). When H 0 holds, ( Λ j(j+,...,p) GGR, n p + j, p j ) Λ j(j+,...,p) 2 2,, Λ j(j+,...,p) Λ j(j+,...,p) ( GGR, p j 2, n p + j 2 ),. We may thus replace Λ as test statistic by G = p Λ j(j+,...,p) Λ j(j+,...,p) or G = p Λ j(j+,...,p) Λ j(j+,...,p), both of which, under H 0 in (4.), have the distribution of the product of independent gamma-ratio rom variables with parameters k j = β j = r j = (n p + j)/2, r 2j = (p j)/2, r j = (p j)/2, r 2j = (n p + j)/2, (j =,...,p ), respectively Rom effects models with balanced cross-nesting. Let us assume that there are L groups with u,...,u L factors. The first factors in the groups will have a l () levels, if u l >, the h-th factor in the l-th group will have a l (h) levels nested inside each level of the preceding factor (h =2,...,u l ). There will be c l (h) = h k= a l(h) level combinations of the first h factors in the l-th group, each nesting b l (h) =c l (u l )/c l (h) level

21 Product ratio of gamma-ratio rom variables 24 combinations of the remaining factors (h =,...,u l ; l =,...,L). We also take c l ( ) = c l (0) =, l =,...,L. Since when a factor is nested in another they do not interact, the variance components will correspond to sets of factors belonging to different groups. These sets of factors are indicated by the vectors h with components h l = 0,...,u l, l =,...,L. If h l = 0, no factor is taken from the l-th group, otherwise h l will be the index of the factor taken in that group. Let Γ represent the set of vectors h σ 2 (h) the variance component for the set of factors corresponding to h. If r observations are taken for each treatment, the sum S(h) of squares of the effects or interactions indicated by h will be (see Fonseca et al., 2003) the product of γ(h) =σ 2 + b(k) σ 2 (k), h Γ, k:h k where b(k)=r k l= b l(k l ), a central chi-square with g(k)= L l= {c l(k l ) c l (k l )} degrees of freedom. When only one component, say h l,ofh, is less than the corresponding bound u l,leth + be the vector obtained increasing that component by. Then γ(h) =γ(h + )+b(h) σ 2 (h), we may use to test F(h) = g(h+ ) g(h) S(h) S(h + ), H 0 (h) : σ 2 (h) =0. We point out that F(h) will be the product of + σ 2 (h)/γ(h + ) a variable having a central F distribution with g(h) g(h + ) degrees of freedom. The S(h), h Γ, are independent (Fonseca et al., 2003). Thus if the pairs (h j,h + j ), j =,...,d, are formed by distinct vectors, we may use to test d F = F(h j ), H 0 : σ 2 (h )= = σ 2 (h d )=0.

22 242 Carlos A. Coelho João T. Mexia This test statistic, when H 0 holds, will be the product of d independent variables with central F distributions Combining independent F tests in meta-analysis. An emergent problem in meta-analysis is how to combine independent F tests (Khuri et al., 998). One possible way to do so is by considering an overall test statistic, for the whole meta-analysis being considered, which will be the product of the elementary F statistics for each single analysis. By using this procedure, we may also obtain the distribution of the overall statistic under the alternative hypothesis, which will be the distribution of the product of non-central F distributions. This distribution may be derived from the distributions obtained in this manuscript will be the subject of a future work. 4.4 A test for equality of two generalized variances. Let us suppose that X N p (µ, Σ ) X 2 N p (µ 2, Σ 2 ), that we want to test the hypothesis H 0 : Σ = Σ 2, (4.4) based on two independent samples of sizes n n 2 of X X 2 respectively. Taking the work of Cao (2006, Chap. 4, sec. 4., 4.2) as a basis, we may use the statistic T = S S 2, (4.5) where S S 2 are either the sample variance-covariance matrices or the maximum likelihood estimators of Σ Σ 2, based on the two independent rom samples of sizes n n 2 respectively of X X 2. Alternatively, S S 2 may yet be taken as the matrices of squares cross-products of the deviations from the sample means. A test of size α for H 0 in (4.4) may then be constructed by rejecting H 0 in (4.4) if, for the statistic T in (4.5), T calc >T α/2 or T calc <T α/2, where T α/2 T α/2 represent respectively the α/2 the α/2 quantiles of the statistic T in (4.5). Taking S S 2 as the maximum likelihood estimators of Σ Σ 2, we will have S W p ( n, n Σ ) S 2 W p ( n 2, n 2 Σ 2 ),

23 with where Product ratio of gamma-ratio rom variables 243 S Σ n p W j S 2 Σ 2 n 2 p Y j, W j χ 2 n j Y j χ 2 n 2 j (j =,...,p) are all independent. Thus, under H 0 in (4.4), the distribution of the statistic T in (4.5) is the same as the distribution of ( ) p p n W p j ) p = W p j p p Y j Y j = Z j, ( n2 where ( Wj n j Γ, n ) 2 2 Y j ( n2 j Γ, n ) (j =,...,p), i.e., under H 0 in (4.4), the exact distribution of the statistic T in (4.5) is the product of p r.v. s ( n Z j GGR, n j, n ) 2 j, (j =,...,p). n A Few Simulations Numerical Studies for the Distributions Obtained We will consider in this section a few numerical studies, together with the results from simulations in order to show the correct behaviour of the distributions obtained the correctness of the corresponding expressions. To show the agreement between the simulated data the distributions obtained, simultaneous plots of the histograms of relative frequencies (simple cumulative) the exact p.d.f. s c.d.f. s are shown. The simulations were carried out by generating five vectors, each with 0 6 romly generated values of the distribution under study. These vectors were then ordered their average used to compute the simulated quantiles plot the relative frequency histograms. We should note that although for any value of n IN, the expressions for the p.d.f. s the c.d.f. s in (3.3), (3.4), (3.6) (3.7) yield proper p.d.f. s c.d.f. s, we need to consider sufficiently high values of n IN in

24 244 Carlos A. Coelho João T. Mexia order that the p.d.f. s the c.d.f. s in (3.3) (3.4) or (3.6) (3.7) converge to the exact distribution of the desired product of GGR r.v. s. All simulations computations were done with the software Mathematica, from Wolfram Research, version The test of independence in a set of p variables. The case presented in this subsection corresponds to the example of application discussed in subsection 4.. For p = 5 N = 0, according to what was exposed in subsection 4., if we want to test the independence of the p = 5 variables, we may, in this case, use the test statistic where G = G G 2 G 3 G 4, (5.) G GGR(, 5/2, 2, ), G 2 GGR(, 6/2, 3/2, ), G 3 GGR(, 7/2,, ), G 4 GGR(, 8/2, /2, ), (5.2) so that, using the result that states that if X χ 2 m X 2 χ 2 m are two independent r.v. s, then ( ) χ 2 2 X X 2 2m 2, 4 we may also write G = F F2, (5.3) where F GGR(, 0/2, 2/2, /2) F2 GGR(, 4/2, 6/2, /2). (5.4) In order to obtain the exact distribution of the statistic G, we may then either use (5.) (5.2) or (5.3) (5.4). In Figure 5., we may observe the overlayed plots of the histograms of relative frequencies generated from the simulated data, obtained from (5.2), the exact p.d.f. c.d.f. for the statistic G in (5.), for n = 00, where we may observe the very long tail that the distribution has in this case.

25 Product ratio of gamma-ratio rom variables (a) (b) Figure 5.. Overlayed plots of the histograms of relative frequencies generated from simulated data the exact p.d.f. c.d.f. for n = 00 for the statistic in (5.): (a) histogram of relative frequencies exact p.d.f.; (b) histogram of cumulative relative frequencies exact c.d.f. In Table 5., we have the simulated 0.90, quantiles for the statistic G in (5.), together with the quantiles computed from different truncations of the exact distributions corresponding to (5.)-(5.2) (5.3)- (5.4). Table 5.. Quantiles q (simulated from different truncations of the exact distribution) for the statistic G in (5.) From the exact distribution From the exact distribution in (5.), (5.2) with n = in (5.3), (5.4) with n = q simulat We may note that although, as expected, both exact distributions have quantiles that converge to the simulated ones as n, the performance of the truncations with n =50n = 00 is rather poor, with the truncations of the exact distribution based on (5.3) (5.4) exhibiting a much worse behaviour. The exact distribution of G based on (5.3) (5.4) is indeed simpler than the one based on (5.) (5.2) since it only involves two r.v. s instead of four, as such it is also much easier quicker to compute. But, on the other h, it seems to need twice the number of terms to attain a similar accuracy as the one exhibited by the distribution based on (5.) (5.2). Although this worse performance of the exact distribution based on the product of two GGR r.v. s may come out as a surprise, in the end

26 246 Carlos A. Coelho João T. Mexia it seems that the result points towards somewhat similar number of terms being necessary while one uses either one of the two exact distributions. Similar conclusions may be drawn from Table 5.2, where we may analyse the values of the relative errors for the quantiles obtained from the truncations of the exact distributions used in Table 5.. Now that we do not have the exact quantiles, the relative errors are computed relative to the simulated quantiles as (approximate simulated)/simulated. Once again, it is interesting to note the almost linear relation between the values of the relative error the value of n used for both exact distributions. Table 5.2. Relative error for quantiles obtained from different truncations of the exact distributions of the statistic G in (5.) (5.3) Value of n in the exact distribution Value of n in the exact distribution in (5.) (5.2) in (5.3) (5.4) Quant The alternative approach is based on using the statistic G = G G 2G 3G 4, (5.5) where G GGR(, 2, 5/2, ), G 2 GGR(, 3/2, 6/2, ), G 3 GGR(,, 7/2, ), G 4 GGR(, /2, 8/2, ) (5.6) or, G = F F 2, (5.7) where F GGR(, 2/2, 0/2, /2) F2 GGR(, 6/2, 4/2, /2). (5.8) In Figure 5.2, we have the overlayed plots of the histograms of relative frequencies generated from the simulated data, obtained from (5.6), the exact p.d.f. c.d.f. for the statistic G in (5.5), for n = 00.

27 Product ratio of gamma-ratio rom variables (a) (b) Figure 5.2. Overlayed plots of the histograms of relative frequencies generated from simulated data the exact p.d.f. c.d.f. for n = 00 for the statistic in (5.5): (a) histogram of relative frequencies exact p.d.f.; (b) histogram of cumulative relative frequencies exact c.d.f.. In Table 5.3, we have the simulated 0.90, quantiles corresponding quantiles from different truncations of the exact distributions derived from (5.6) (5.8), in Table 5.4, the relative errors of these quantiles computed taking as reference the values of the simulated quantiles. Table 5.3. Quantiles q (simulated from different truncations of the exact distribution) for the statistic G in (5.5) (5.7) From the exact distribution From the exact distribution in (5.5), (5.6) with n = in (5.7), (5.8) with n = q simulat Table 5.4. Relative errors for quantiles obtained from different truncations of the exact distributions of the statistic G in (5.5) (5.7) Value of n in the exact distribution Value of n in the exact distribution in (5.5), (5.6) in (5.7), (5.8) Quant We may observe in Table 5.4 that the relative errors are, for a given value of n, generally smaller when using the statistic G, with a noticeable small value of the relative error for the 0.95 quantiles for n = 200.

28 248 Carlos A. Coelho João T. Mexia (a) (b) Figure 5.3. Overlayed plots of the histograms of relative frequencies generated from simulated data the exact p.d.f. c.d.f. for n = 300 for the statistic in (5.9): (a) histogram of relative frequencies exact p.d.f.; (b) histogram of cumulative relative frequencies exact c.d.f Simultaneous tests for variance components in rom effects models with balanced cross-nesting. Let us assume we have three groups of factors, the first the third of which have a single factor the second with five factors sequentially nested. Let us further suppose that all factors have two levels rom effects. When we want to test the simultaneous nullity of the variance components associated with the factor in the first group with its interaction with the second fourth factors in the second group, we have to use the statistic F = F F 2 F 3, (5.9) where F = F 2 = 4 2 F 3 = 6 8 ([ ]) S 0 ([ S ([ ]) S 2 ([ S 3 ([ ]) S 4 ([ S 5 ]) F, GGR (, 2, 2, ) ]) F 2,4 GGR ( 2 4, 2 2, 4 2, ) ]) F 8,6 GGR ( 8 6, 8 2, 6 2, ). Figure 5.3 shows the good graphical agreement between the relative frequency histograms generated from the simulated data the exact p.d.f.

29 Product ratio of gamma-ratio rom variables 249 c.d.f., while in Tables we may see once again that in order to be able to achieve a reasonable good accuracy for the computation of quantiles we may need quite a large number of terms in the exact distribution. Table 5.5. Quantiles (simulated from different truncations of the exact distribution) for the statistic F in (5.9) From the exact distribution with n = Quant. Simulat In Table 5.6, we report the values of the relative errors for the quantiles obtained from the truncations of the exact distribution used in Table 5.5. Once again, we may note the almost linear decay of the values of the relative errors for increasing values of n. Only for the quantile 0.95, this rate seems to be slightly higher, what may indeed be due to the fact that the value of the simulated quantile, relative to which the relative errors are computed, may be a bit higher than what it should be. Table 5.6. Relative error for quantiles obtained from different truncations of the exact distribution of the statistic F in (5.9) Value of n in the exact distribution Quant Relations with Other Distributions We should emphasize that the results obtained may be easily directly generalized to the case where we are interested in the distribution of the r.v. Z = n γ j Y α j j, where γ j IR +, α j IR\{0} Y j are independent r.v. s with GGR distributions, since it is straightforward to show that if Y j GGR(k j,r j,r 2j,β j ), (6.)

30 250 Carlos A. Coelho João T. Mexia then γ j Y α j j GGR k j γ β j/α j j,r j,r 2j, β j. α j The distributions obtained in section 3 may also be seen as the distributions of the product of independent beta prime r.v. s. Recall that if the r.v. X has a stard beta distribution, the distribution of either ( X)/X or X/( X) is then usually called a stard beta prime or beta second kind distribution. However, we should note that the distributions of either ( X)/X or X/( X) are then only particular GGR distributions. Actually, it is easy to show that if Y j GGR(k j,r j,r 2j,β j ) with k j = β j =, then the r.v. s /( + Y j )Y j /( + Y j )have stard beta distributions with parameters r 2j r j or r j r 2j, respectively, while for general k j > 0 general β j IR\{0} the r.v. X j = Y j /(+Y j ) has what we call a generalized beta distribution with p.d.f. f Xj (x) = β j k r j ( ) j x rj r 2j +k j ( x) β jr j x β jr j, B(r j,r 2j ) x which clearly reduces to the stard beta p.d.f. for k j = β j =, while the r.v. X j =/( + Y j ) has of course a similar p.d.f. with r j r 2j swapped. For this reason, the distribution of Y j is also called, for k j = β j =, a beta prime distribution for general k j β j a generalized beta prime distribution. Thus, for the distribution in (6.), we may also say that Y j has a generalized beta prime distribution thus, the distributions obtained in section 3 are also the distributions of the product of independent generalized beta prime r.v. s. Clearly, if in (6.) we have r j = m j /2, r 2j = n j /2, with m j,n j IN, k = m j /n j β j =, the r.v. s Y j will have F distributions with m j n j degrees of freedom, thus the results in section 3 will then give the distribution of the product of independent F distributed r.v. s. Also, if in (6.) we have r j =/2, r 2j = n j /2, k j =/n j β j =2,with n j IN, wehavey j with the so-called folded T distribution, which is the distribution of a r.v. Y j = T j, where T j has a student s t distribution with n j degrees of freedom. If instead we take r 2j =/2 k j =, we will have Y j with the so-called folded Cauchy distribution, that is the distribution

31 Product ratio of gamma-ratio rom variables 25 of the absolute value of a r.v. with a stard Cauchy distribution, or a student s t distribution with only degree of freedom. In both cases, once again, the results in section 3 may be readily applied. Since, contrary to what is commonly done, we considered the power parameters as real not necessarily positive (only non-zero), by allowing them to be negative, the results obtained may be directly extended to the distribution of the ratio of two independent GGR rom variables or the distribution of the ratio of two independent products of independent GGR rom variables. In order to obtain the distribution of the ratio of two independent GGR rom variables, one simply has to consider m = 2 in (3.2), taking then for the rom variable in the denominator the negative of its power parameter. The distribution for the ratio of two independent products of independent GGR rom variables may then be obtained by considering the distribution of the product of the whole set of rom variables, taking the power parameters for the rom variables in the denominator with a minus sign. As a by-product, in subsections , we also obtain closed form representations, not involving any infinite series or unsolved integrals, for the distribution of the difference of two independent sums of a finite number of exponential rom variables with all different rate parameters or for the distribution of the difference of two independent sums of gamma rom variables with all different rate parameters integer shape parameters, under the form of particular mixtures of either exponential or gamma distributions, according to the case. Also, if we consider the exponential of a gamma rom variable with integer shape parameter as a generalized Pareto distribution, then the distribution of the rom variable W in subsection 2.3 is the distribution of the ratio of two independent products of Pareto distributions, while the distribution of the same rom variable in subsection 2.4 is the distribution of the ratio of two independent products of generalized Pareto distributions, expressed as a particular mixture of Pareto inverted Pareto distributions. 7 Conclusions Final Remarks Given the form of the exact distributions obtained for the product of GGR r.v. s the problems related to the need for the evaluation of a large number of terms in the series in order to obtain the desired precision, the development of near-exact distributions seems to be a much desirable

32 252 Carlos A. Coelho João T. Mexia goal. Such near-exact distributions are indeed not too hard to obtain, being not treated in this paper due to length limitations. The simple truncation of the series obtained gives indeed unsatisfactory results, namely in terms of the c.d.f. the quantiles, owing to the fact that in this case the weights do not add up to, preventing the c.d.f. from reaching this value of. Much better results may be obtained if we consider near-exact distributions, based on the concept of keeping the major part of the exact characteristic function unchanged approaching the remaining part by an asymptotic result (Coelho, 2003, 2004). Given the form obtained for the exact distributions, such an approach would lead us to consider, for example, the truncation of the infinite series obtained, coupled with one or two more terms, which would at the same time enable the weights to add up to the first few moments to match the exact ones. Yet, the extension of the results presented in this paper to non-central distributions is almost straightforward, as it only requires consideration of the mixtures with Poisson weights that would account for the non-centrality. Once again, this topic is not addressed in this paper as the paper is already quite long. Appendix The facts that in (2.2)-(2.3), thus also in (2.4)-(2.7) (2.8)-(2.9), we may take p, also in (2.26)-(2.3) we may take p p 2, with no need to worry about the convergence of the series involved, as well as about the convergence of the infinite products in (3.) (3.2), are justified as follows. If X i Γ(r j,λ j ), i =, 2,..., then the characteristic function of is with λ r j λ r j j (λ j it) r j Z = = lim j (λ j it) r j = lim X j p p p λ r j p λ r j j (λ j it) r j, j (λ j it) r j, (A.)

33 Product ratio of gamma-ratio rom variables 253 not only because λr j j (λ j it) r j is a characteristic function but also because λ r j j (λ j it) r j p λ r j j (λ j it) r j (A.2) for any p <, because for any j, we clearly have λ r j j (λ j it) r j. But then, let f Z (z; r,λ,p)= p e itz 2π represent the p.d.f. of the rom variable λ r j j (λ j it) r j dt Z = X j, let f Z (z; ) = lim p Z (z; r,λ; p) represent the p.d.f. of the rom variable Z in (A.). The fact that f Z (z; ) is a legitimate p.d.f. follows directly from (A.2) the dominated convergence theorem, since thus f Z (z; ) = lim e itz p 2π = 2π e itz p λ r j j (λ j it) r j dt λ r j j (λ j it) r j dt (where the integral is uniformly convergent given the result in (A.2)). If all the shape parameters r j (j =, 2,...) are integer, the p.d.f f Z (z; ) is the p.d.f. of a GIG distribution with infinite depth, which shows that in expressions (2.4) through (2.20) we may take p. Similar arguments could then be used for the distribution of the difference of the two sums of infinitely many exponential rom variables considered after expression (2.23), or the distribution of the difference of the two sums of

34 254 Carlos A. Coelho João T. Mexia infinitely many gamma rom variables all with integer shape parameters, considered at the end of Section 2, both of these distributions being the distribution of the difference of two GIG distributions with infinite depth. Acknowledgements. The authors wish to thank the remarks suggestions made by the referees editors which improved the presentation of the paper. The reviewers suggested the need for Sections 4 5, which help in understing the scope, usefulness functioning of the distributions presented in the paper. The authors also want to thank Professor Thomas Mathew of the University of Maryl, who, as the adviser, brought to our attention made available to us the material in Cao (2006). References Ahlfors, L.V. (979). Complex Analysis : An Introduction to the Theory of Analytic functions of One Complex Variable, 3rd edition, McGraw-Hill, London. Anderson, T.W. (984). An Introduction to Multivariate Statistical Analysis, 2ndedition, Wiley, New York. Cao, L. (2006). The Assessment of Multivariate Bioequivalence. Ph.D. Thesis, Department of Mathematics Statistics, The University of Maryl, Baltimore County, MD. Coelho, C.A. (992). The Generalized Canonical Analysis. Ph.D. Thesis, The University of Michigan, Ann Arbor, MI. Coelho, C.A. (998). The generalized integer gamma distribution a basis for distributions in multivariate statistics, J. Multivariate Anal., 64, Coelho, C.A. (2003). A generalized integer gamma distribution as an asymptotic replacement for a logbeta distribution & applications, Amer. J. Math. Management Sci., 23, Coelho, C.A. (2004). The generalized near-integer gamma distribution: a basis for near-exact approximations to the distribution of statistics which are the product of an odd number of independent beta rom variables, J. Multivariate Anal., 89, Coelho, C.A. Mexia, J.T. (2006). On the exact near-exact distributions of statistics used in generalized F tests. Amer. J. Math. Management Sci. (submitted). Fonseca, M., Mexia, J.T. Zmyslony, R. (2003). Estimators tests for variance components in cross nested orthogonal designs, Discuss. Math., Probab. Stat., 23, Khuri, A.I., Mathew, T. Sinha, B.K. (998). Statistical Tests for Mixed Linear Models, Wiley, New York. Lang, S. (999). Complex Analysis, 4th edition, Springer-Verlag, New York. Pham-Gia, T. Turkkan, N. (2002). Operations on the generalized-f variables applications. Statistics, 36,

35 Product ratio of gamma-ratio rom variables 255 Provost, S.B. (986). On the distribution of some test statistics connected with the multivariate linear functional relationship model. Commun. Statist. Theory Methods, 5, Shah, M.C. Rathie, P.N. (974). Exact distribution of product of generalized F -variates. Canad. J. Statist., 2, Carlos A. Coelho João T. Mexia Department of Mathematics Faculty of Science Technology The New University of Lisbon caparica, Portugal cmac@fct.unl.pt jtm@fct.unl.pt Paper received November 2005; revised May 2007.