## Probability Surveys

### Characterizations of GIG laws: A survey

#### Abstract

Several characterizations of the Generalized Inverse Gaussian (GIG) distribution on the positive real line have been proposed in the literature, especially over the past two decades. These characterization theorems are surveyed, and two new characterizations are established, one based on maximum likelihood estimation and the other is a Stein characterization.

#### Article information

Source
Probab. Surveys, Volume 11 (2014), 161-176.

Dates
First available in Project Euclid: 17 July 2014

https://projecteuclid.org/euclid.ps/1405603141

Digital Object Identifier
doi:10.1214/13-PS227

Mathematical Reviews number (MathSciNet)
MR3264557

Zentralblatt MATH identifier
1296.60027

#### Citation

Koudou, Angelo Efoévi; Ley, Christophe. Characterizations of GIG laws: A survey. Probab. Surveys 11 (2014), 161--176. doi:10.1214/13-PS227. https://projecteuclid.org/euclid.ps/1405603141

#### References

• [1] Barndorff-Nielsen, O. E. and Halgreen, C. (1977). Infinite divisibility of the Hyperbolic and generalized inverse Gaussian distribution. Z. Wahrsch. Verw. Geb. 38, 309–312.
• [2] Barndorff-Nielsen, O. E. and Koudou, A. E. (1998). Trees with random conductivities and the (reciprocal) inverse Gaussian distribution. Adv. Appl. Probab. 30, 409–424.
• [3] Bernadac, E. (1995). Random continued fractions and inverse Gaussian distribution on a symmetric cone. J. Theor. Probab. 8, 221–260.
• [4] Brown, T. and Phillips, M. (1999). Negative binomial approximation with Stein’s method. Methodol. Comput. Appl. Probab. 1, 407–421.
• [5] Chatterjee, S., Fulman, S. and Röllin, A. (2011). Exponential approximation by Stein’s method and spectral graph theory. ALEA Lat. Am. J. Probab. Math. Stat. 8, 197–223.
• [6] Chebana, F., El Adlouni, S. and Bobée, B. (2010). Mixed estimation methods for Halphen distributions with applications in extreme hydrologic events. Stoch. Environ. Res. Risk Assess. 24, 359–376.
• [7] Chen, L. H. Y. (1975). Poisson approximation for dependent trials. Ann. Probab. 3, 534–545.
• [8] Chhikara, R. S. and Folks, J. L. (1989). The Inverse Gaussian Distribution, Theory, Methodology and Applications. Marcel Dekker Inc, New-York.
• [9] Chou, C.-W. and Huang, J.-W. (2004). On characterizations of the gamma and the generalized inverse Gaussian distributions. Stat. Probab. Lett. 69, 381–388.
• [10] Duerinckx, M., Ley, C. and Swan, Y. (2013). Maximum likelihood characterization of distributions. Bernoulli, to appear.
• [11] Eberlein, E. and Hammerstein, E. A. (2004). Generalized hyperbolic and inverse Gaussian distributions: limiting cases and approximation of processes. Prog. Probab. 58, 221–264.
• [12] Gauss, C. F. (1809). Theoria motus corporum coelestium in sectionibus conicis solem ambientium. Cambridge Library Collection. Cambridge University Press, Cambridge. Reprint of the 1809 original.
• [13] Goldstein, L. and Reinert, G. (2013). Stein’s method for the Beta distribution and the Pòlya-Eggenberger urn. Adv. Appl. Probab., to appear.
• [14] Good, I. J. (1953). The population frequencies of species and the estimation of population parameters. Biometrika 40, 237–260.
• [15] Halgreen, C. (1979). Self-decomposability of the generalized inverse Gaussian and hyperbolic distributions. Z. Wahrsch. verw. Geb. 47, 13–17.
• [16] Halphen, E. (1941). Sur un nouveau type de courbe de fréquence. Comptes Rendus de l’Académie des Sciences 213, 633–635. Published under the name of “Dugué” due to war constraints.
• [17] Hürlimann, W. (1998). On the characterization of maximum likelihood estimators for location-scale families. Comm. Statist. Theory Methods 27, 495–508.
• [18] Iyengar, S. and Liao, Q. (1997). Modeling neural activity using the generalized inverse Gaussian distribution. Biol. Cybern. 77, 289–295.
• [19] Jørgensen, B. (1982). Statistical Properties of the Generalized Inverse Gaussian Distribution. Springer-Verlag, Heidelberg.
• [20] Kagan, A. M., Linnik, Y. V. and Rao, C. R. (1973). Characterization Problems in Mathematical Statistics. Wiley, New York.
• [21] Khatri, C. G. (1962). A characterisation of the inverse Gaussian distribution. Ann. Math. Statist. 33, 800–803.
• [22] Kawamura, T. and Kōsei, I. (2003). Characterisations of the distributions of power inverse Gaussian and others based on the entropy maximisation principle. J. Japan Statist. Soc. 33, 95–104.
• [23] Koudou, A. E. (2006). A link between the Matsumoto-Yor property and an independence property on trees. Stat. Probab. Lett. 76, 1097–1101.
• [24] Koudou, A. E. and Ley, C. (2014). Efficiency combined with simplicity: new testing procedures for Generalized Inverse Gaussian models. Test, to appear. DOI: 10.1007/s11749-014-0378-2.
• [25] Koudou, A. E. and Vallois, P. (2011). Which distributions have the Matsumoto-Yor property? Electron. Commun. Prob. 16, 556–566.
• [26] Koudou, A. E. and Vallois, P. (2012). Independence properties of the Matsumoto-Yor type. Bernoulli 18, 119–136.
• [27] Le Cam, L. and Yang, G. L. (2000). Asymptotics in statistics. Some basic concepts. 2nd ed. Springer-Verlag, New York.
• [28] Letac, G. and Mora, M. (1990). Natural real exponential families with cubic variance functions. Ann. Statist. 18, 1–37.
• [29] Letac, G. and Seshadri, V. (1983). A characterization of the generalized inverse Gaussian distribution by continued fractions. Z. Wahrsch. Verw. Geb. 62, 485–489.
• [30] Letac, G. and Seshadri, V. (1985). On Khatri’s characterization of the inverse Gaussian distribution. Can. J. Stat. 13, 249–252.
• [31] Letac, G. and Wesołowski, J. (2000). An independence property for the product of GIG and gamma laws. Ann. Probab. 28, 1371–1383.
• [32] Ley, C. and Swan, Y. (2013). Stein’s density approach and information inequalities. Electron. Comm. Probab. 18, 1–14.
• [33] Loh, W. L. (1992). Stein’s method and multinomial approximation. Ann. Appl. Probab. 2, 536–554.
• [34] Luk, H.-M. (1994). Stein’s method for the gamma distribution and related statistical applications. Ph.D. thesis. University of Southern California. Los Angeles, USA.
• [35] Lukacs, E. (1956). Characterization of populations by properties of suitable statistics. Proc. Third Berkeley Symp. on Math. Statist. and Prob., Vol. 2 (Univ. of Calif. Press), 195–214.
• [36] Madan, D., Roynette, B. and Yor, M. (2008). Unifying Black-Scholes type formulae which involve Brownian last passage times up to a finite horizon. Asia-Pacific Finan. Markets 15, 97–115.
• [37] Marshall, A. W. and Olkin, I. (1993). Maximum likelihood characterizations of distributions. Statist. Sinica 3, 157–171.
• [38] Massam, H. and Wesołowski, J. (2004). The Matsumoto-Yor property on trees. Bernoulli 10, 685–700.
• [39] Massam, H. and Wesołowski, J. (2006). The Matsumoto-Yor property and the structure of the Wishart distribution. J. Multivariate Anal. 97, 103–123.
• [40] Matsumoto, H. and Yor, M. (2001). An analogue of Pitman’s $2M-X$ theorem for exponential Wiener functional, Part II: the role of the generalized inverse Gaussian laws. Nagoya Math. J. 162, 65–86.
• [41] Matsumoto, H. and Yor, M. (2003). Interpretation via Brownian motion of some independence properties between GIG and Gamma variables. Stat. Probab. Lett. 61, 253–259.
• [42] Mudholkar, S. M. and Tian, L. (2002). An entropy characterization of the inverse Gaussian distribution and related goodness-of-fit test. J. Stat. Plan. Infer. 102, 211–221.
• [43] Peköz, E. (1996). Stein’s method for geometric approximation. J. Appl. Probab. 33, 707–713.
• [44] Perreault, L., Bobée, B. and Rasmussen, P. F. (1999a). Halphen distribution system. I: Mathematical and statistical properties. J. Hydrol. Eng. 4, 189–199.
• [45] Perreault, L., Bobée, B. and Rasmussen, P. F. (1999b). Halphen distribution system. II: Parameter and quantile estimation. J. Hydrol. Eng. 4, 200–208.
• [46] Poincaré, H. (1912). Calcul des probabilités. Carré-Naud, Paris.
• [47] Pusz, J. (1997). Regressional characterization of the Generalized inverse Gaussian population. Ann. Inst. Statist. Math. 49, 315–319.
• [48] Ross, N. (2011). Fundamentals of Stein’s method. Probab. Surv. 8, 210–293.
• [49] Seshadri, V. (1993). The Inverse Gaussian Distribution, a case study in exponential families. Oxford Science Publications.
• [50] Seshadri, V. (1999). The Inverse Gaussian Distribution. Springer-Verlag, New York.
• [51] Seshadri, V. and Wesołowski, J. (2001). Mutual characterizations of the gamma and the generalized inverse Gaussian laws by constancy of regression. Sankhya Ser. A 63, 107–112.
• [52] Seshadri, V. and Wesołowski, J. (2004). Martingales defined by reciprocals of sums and related characterizations. Comm. Statist. Theory Methods 33, 2993–3007.
• [53] Shannon, C. E. (1949). The Mathematical Theory of Communication. Wiley, New York.
• [54] Sichel, H. S. (1974). On a distribution representing sentence-length in written prose. J. R. Stat. Soc. Ser. A 137, 25–34.
• [55] Sichel, H. S. (1975). On a distribution law for word frequencies. J. Am. Stat. Assoc. 70, 542–547.
• [56] Stein, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Vol 2, pp. 586–602). Berkeley: University of California Press.
• [57] Stein, C., Diaconis, P., Holmes, S. and Reinert, G. (2004). Use of exchangeable pairs in the analysis of simulations. In Persi Diaconis and Susan Holmes, editors, Stein’s method: expository lectures and applications, volume 46 of IMS Lecture Notes Monogr. Ser, pages 1–26. Beachwood, Ohio, USA: Institute of Mathematical Statistics.
• [58] Teicher, H. (1961). Maximum likelihood characterization of distributions. Ann. Math. Statist. 32, 1214–1222.
• [59] Vallois, P. (1991). La loi Gaussienne inverse généralisée comme premier ou dernier temps de passage de diffusion. Bull. Sc. Math., $2^{e}$ Série, 115, 301–368.
• [60] Vasicek, O. (1976). A test for normality based on sample entropy. J. R. Stat. Soc. Ser. B 38, 54–59.
• [61] Wesołowski, J. (2002). The Matsumoto-Yor independence property for GIG and Gamma laws, revisited. Math. Proc. Camb. Philos. Soc. 133, 153–161.
• [62] Wesołowski, J. and Witkowski, P. (2007). Hitting times of Brownian motion and the Matsumoto-Yor property on trees. Stoch. Proc. Appl. 117, 1303–1315.