On characterizations of the gamma and generalized inverse Gaussian distributions☆
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 definedwhere 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,p∈R.
- (ii)
Class II: a>0,b=0,p>0.
- (iii)
Class III: a=0,b>0,p<0.
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,and the GIG distributionwhere p,a,b>0 and K−p is a modified Bessel function withNote 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 μ−p,a,b is the same as μ−p,2a,2b in SW.
Given Y∼Γ(p,a), SW characterized that X∼μ−p,a,b under the assumptions that for r=0 or −1, if r>−p,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 Y∼Γ(p,a) if X∼μ−p,a,b, and (3) holds for r=0 or −1, if r>−p, with . 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) andhold 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,andwhere s⩽0. Note that if Y∼Γ(p,a), then for every r>−p,
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 Y∼Γ(p,a) and for some fixed . Assume that (3) holds for some constant cr. Then cr>0 and X∼μ−p,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 towhich in turns implies
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 X∼μ−p,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 Y∼Γ(p,a). Proof First define W as in Theorem 1. From , it can be seen that W∼μ−(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(X−r−2),E(X−r),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 X∼μ−p,a,b and Y∼Γ(p,a), where p=cr/(cr+1−cr)−r>0 and b=1/(cr+1−cr)>0.
Proof
First , yield that cr+1,cr>0 immediately. Next from (4) we havewhich in turn
Acknowledgements
We are grateful to Professor Wesolowski for many helpful discussions, and sending us the preprint of Wesolowski (2002).
References (24)
- et al.
Interpretation via Brownian motion of some independence properties between GIG and gamma variables
Statist. Probab. Lett
(2003) Regression versions of Lukacs type characterizations for the bivariate gamma distribution
J. Appl. Statist. Sci
(2002)- et al.
Three dual regression schemes for the Lukacs theorem
Metrika
(2002) - et al.
The Lukacs–Olkin–Rubin theorem without invariance of the “quotient”
Studia Math
(2002) - et al.
A characterization of some distributions by conditional moments
Ann. Math. Statist
(1965) - et al.
The Lukacs–Olkin–Rubin characterization of Wishart distributions on symmetric cones
Ann. Statist
(1996) - et al.
The Inverse Gaussian Distribution, Theory, Methodology and Applications
(1989) - et al.
Characterizations of the gamma distribution via conditional moments
Sankhyā
(2003) - Chou, C.W., Huang, W.J., 2004. A note on characterizations of the bivariate gamma distribution. J. Statist. Plann....
- et al.
On characterizations of the gamma distribution
Sankhyā A
(1969)
On a study of renewal process connected with certain conditional moments
Sankhyā A
Quadratic and inverse regressions for Wishart distributions
Ann. Statist
Cited by (22)
Improved estimators for the rate parameter of gamma model using asymptotic properties
2021, HeliyonCitation Excerpt :Over the last few decades, the gamma distribution has become one of the most important techniques for modeling life-testing situations, Chou and Huang (2003). In spite of many uses of the gamma distribution, there have been very few distributional assessment procedures developed, Chou and Huang (2004). The two-parameter gamma distribution has been used widely in many studies including reliability studies, survival analysis, and in many applications particularly when data are complete or incomplete.
Change of measure technique in characterizations of the gamma and Kummer distributions
2018, Journal of Mathematical Analysis and ApplicationsCitation Excerpt :Two important consequences of Theorem 2.1 will be stated in the following theorems. Similar extensions of the Lukacs and the Matsumoto–Yor properties were given in [8] and [9], respectively. However these authors, instead of the change of measure method, which was applied above, used the standard but more cumbersome approach leading to differential equations for characteristic functions.
On the Matsumoto-Yor type regression characterization of the gamma and Kummer distributions
2015, Statistics and Probability LettersAn alternative inverse Gaussian distribution
2009, Mathematics and Computers in SimulationMultivariate Matsumoto-Yor property is rather restrictive
2007, Journal of Statistical Planning and InferenceSimulation Study of Estimators of the Gamma Rate Parameter Using MLE as a Baseline Estimator
2024, Journal of the Indian Society for Probability and Statistics
- ☆
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.