On characterizations of the gamma and generalized inverse Gaussian distributions

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Abstract

Given two independent non-degenerate positive random variables X and Y, Letac and Wesolowski (Ann. Probab. 28 (2000) 1371) proved that U=(X+Y)−1 and V=X−1−(X+Y)−1 are independent if and only if X and Y are generalized inverse Gaussian (GIG) and gamma distributed, respectively. Note that X=(U+V)−1 and Y=U−1−(U+V)−1. This interesting transformation between (X,Y) and (U,V) preserves a bivariate probability measure which is a product of GIG and gamma distributions.

In this work, characterizations of the GIG and gamma distributions through the constancy of regressions of Vr on U are considered.

Introduction

Lukacs (1955) characterized two independent non-degenerate positive random variables to be gamma distributed by the independence of their quotient and sum. Since then, there are many further investigations. Among others the following are some basic directions: (i) Weakening the independence condition to constancy of regressions—see Bolger and Harkness (1965), Hall and Simons (1969), Wesolowski (1990), Li et al. (1994) Huang and Su (1997), Bobecka and Wesolowski (2002a), Chou and Huang (2003). (ii) considering the renewal process—see Wesolowski (1989), Li et al. (1994), Huang and Su (1997), Chou and Huang (2003). (iii) Considering the bivariate cases—see Wang (1981), Bobecka (2002), Pusz (2002), Chou and Huang (2004). (iv) Considering the matrix variates—see Olkin and Rubin (1962), Casalis and Letac (1996), Letac and Massam (1998), Bobecka and Wesolowski (2002b).

Letac and Wesolowski (2000) (LW in the sequel) gave a similar characterization of generalized inverse Gaussian (GIG) and gamma distributions via independence condition: given two independent non-degenerate positive random variables X and Y, if U=(X+Y)−1 and V=X−1−(X+Y)−1 are independent, then X is GIG distributed and Y is gamma distributed. The readers may refer to Pusz (1997) and Matsumoto and Yor (2003) for some related works of GIG distribution. Again, there are at least two directions developed after LW: (i) Weakening the independence condition to constancy of regressions (see Seshadri and Wesolowski (2001) (SW in the sequel), and Wesolowski (2002)). (ii) Considering the matrix variates (see Wesolowski, 2002).

The GIG distribution μp,a,b is definedμp,a,b(dx)=Cxp−1exp(−ax−b/x)I(0,∞)(x)dx,where C is the norming constant. The family of GIG distribution can be partitioned into the following three classes according to the parameter (p,a,b):

  • (i)

    Class I: a>0,b>0,pR.

  • (ii)

    Class II: a>0,b=0,p>0.

  • (iii)

    Class III: a=0,b>0,p<0.

Note that if X is μp,a,b distributed, then X−1 is μp,b,a distributed. Class I contains the inverse Gaussian (IG) (with p=−12), reciprocal inverse Gaussian (RIG) (p=12), hyperbolic (p=1) and hyperbola (p=0) distributions. Class II is the class of gamma distributions. Class III is the class of reciprocal gamma distributions. For the details of the IG distribution characteristics and various statistical methods, see Chhikara and Folks (1989), Seshadri (1999) and the references therein. A random variable X is RIG (or reciprocal gamma) distributed, if and only if X−1 is IG (or gamma) distributed.

In this work we consider only Class II and a subclass of Class I. More precisely, we consider the gamma distribution Γ(q,c) (i.e. μq,c,0), where q,c>0,γq,c(dy)=cqΓ(q)yq−1e−cyI(0,∞)(y)dy,and the GIG distributionμ−p,a,b(dx)=(a/b)−p/22K−p(2ab)x−p−1exp(−ax−b/x)I(0,∞)(x)dx,where p,a,b>0 and Kp is a modified Bessel function withK−p(z)=1212z−p0up−1exp−u−z24udu.Note that the definition in (2) is the same as that in Wesolowski (2002) while somewhat different from that in SW in parameters a and b. In fact, our μp,a,b is the same as μp,2a,2b in SW.

Given YΓ(p,a), SW characterized that Xμp,a,b under the assumptions that for r=0 or −1, if r>−p,E(Vr+1|U)=crE(Vr|U),holds for some constant cr. In this work we will prove that the above result holds if (3) is true for some fixed real r>−p. On the other hand, SW also characterized that YΓ(p,a) if Xμp,a,b, and (3) holds for r=0 or −1, if r>−p, with c0=E(V),c−1=1/E(V−1). Again, we prove that the result still holds if (3) is true for some r>−p.

Simultaneous characterizations of the distributions of X and Y are considered in Wesolowski (2002), he characterized X to be GIG distributed and Y gamma distributed under the assumption that for r=−1, (3) andE(Vr+2|U)=cr+1E(Vr+1|U),hold for some constants cr and cr+1. Some further extension will be given in this work.

Before going into the details, we define the following transforms, if they exist,fX(s)=E(X−r−1esX),gX(s)=E(X−r−2esX),andhY(s)=E(YresY),where s⩽0. Note that if YΓ(p,a), then for every r>−p,hY(s)=E(Yr)(1−s/a)−(p+r),s⩽0.

Section snippets

Characterization of GIG distribution given that Y is gamma distributed

In this section, we characterize X to be GIG distributed given the distribution of Y is gamma and Eq. (3) holds for some fixed r>−p.

The following theorem generalizes Theorem 1, Theorem 3 of SW.

Theorem 1

Let YΓ(p,a) and for some fixed r>−p,E(X−r−1),E(X−r+1)<∞. Assume that (3) holds for some constant cr. Then cr>0 and Xμp,a,b, where b=(p+r)/cr>0.

Proof

First (3) implies cr>0 immediately. Next from the definitions of U and V, (3) is equivalent toEYX(X+Y)r+1|(X+Y)−1=crEYX(X+Y)r|(X+Y)−1,which in turns impliesE(X

Characterization of gamma distribution given that X is GIG distributed

In Theorem 1, knowing the distribution of Y can characterize the distribution of X. Alternatively, in this section we characterize the distribution of Y when the distribution of X is known.

The following theorem generalizes Theorems 2 and 4 of SW.

Theorem 2

Let Xμp,a,b, with p,a,b>0. Assume that for some fixed r>−p,E(Yr),E(Yr+1)<∞, and (3) holds with cr=(p+r)/b>0. Then YΓ(p,a).

Proof

First define W as in Theorem 1. From , it can be seen that Wμ−(p+r),a,b. Next let FY denote the distribution function of Y,Z

Simultaneous characterization of GIG and gamma distributions given two conditional expectations

In this section, we give a simultaneous characterization of the distributions of X and Y.

The following theorem extends Theorem 1 of Wesolowski (2002).

Theorem 3

Assume that E(Xr−2),E(Xr),E(Yr) and E(Yr+2)<∞ for some fixed r. If (3) and (4) hold for some constants cr and cr+1, then

  • (i)

    cr+1>cr>0;

  • (ii)

    there exists a>0 such that Xμp,a,b and YΓ(p,a), where p=cr/(cr+1cr)−r>0 and b=1/(cr+1cr)>0.

Proof

First , yield that cr+1,cr>0 immediately. Next from (4) we haveE1X1X+YYXr+1|(X+Y)−1=cr+1EYXr+1|(X+Y)−1,which in turn

Acknowledgements

We are grateful to Professor Wesolowski for many helpful discussions, and sending us the preprint of Wesolowski (2002).

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    Support for this research was provided in part by the National Science Council of the Republic of China, Grant No. NSC 91-2118-M390-001.

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