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The Logarithmic Skew-Normal Distributions are Moment-Indeterminate

Part of: Integral transforms, operational calculus Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Gwo Dong Lin  and
Jordan Stoyanov
Gwo Dong Lin*
Affiliation:
Academia Sinica, Taipei
Jordan Stoyanov*
Affiliation:
Newcastle University
*
∗Postal address: Institute of Statistical Science, Academia Sinica, Taipei 11529, Taiwan, Republic of China.
∗∗Postal address: School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK. Email address: jordan.stoyanov@ncl.ac.uk
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Abstract

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We study the class of logarithmic skew-normal (LSN) distributions. They have heavy tails; however, all their moments of positive integer orders are finite. We are interested in the problem of moments for such distributions. We show that the LSN distributions are all nonunique (moment-indeterminate). Moreover, we explicitly describe Stieltjes classes for some LSN distributions; they are families of infinitely many distributions, which are different but have the same moment sequence as a fixed LSN distribution.

Keywords

Logarithmic skew-normal distributionheavy tailfinite momentsproblem of momentsnonuniquenessStieltjes class

MSC classification

Secondary: 60E05: Distributions 44A60: Moment problems
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 3 , September 2009 , pp. 909 - 916
Copyright
Copyright © Applied Probability Trust 2009 

Footnotes

Dedicated to the memory of Professor Chris Heyde (1939-2008).

References

Akhiezer, N. I. (1965). The Classical Moment Problem and Some Related Questions in Analysis. Hafner Publishing, New York.Google Scholar
Arnold, B. C. and Beaver, R. J. (2000). Hidden truncation models. Sankhyā A 62, 23–35.Google Scholar
Arnold, B. C. and Lin, G. D. (2004). Characterizations of the skew-normal and generalized chi distributions. Sankhyā 66, 593–606.Google Scholar
Azzalini, A. (1985). A class of distributions which includes the normal ones. Scand. J. Statist. 12, 171–178.Google Scholar
Azzalini, A., Dal Cappello, T. and Kotz, S. (2003). Log-skew-normal and log-skew-t distributions as models for family income data. J. Income Distribution 11, 12–20.Google Scholar
Azzalini, A. and Dalla Valle, A. (1996). The multivariate skew-normal distribution. Biometrika 83, 715–726.Google Scholar
Chai, H. S. and Bailey, K. R. (2008). Use of log-skew-normal distribution in analysis of continuous data with a discrete component at zero. Statist. Med. 27, 3643–3655.CrossRefGoogle ScholarPubMed
Gradshteyn, I. S. and Ryzhik, I. M. (2000). Tables of Integrals, Series, and Products, 6th edn. Academic Press, San Diego, CA.Google Scholar
Henze, N. (1986). A probabilistic representation of the ‘skew-normal’ distribution. Scand. J. Statist. 13, 271–275.Google Scholar
Heyde, C. C. (1963). On a property of the lognormal distribution. J. R. Statist. Soc. B 25, 392–393.Google Scholar
Lin, G. D. (1997). On the moment problems. Statist. Prob. Lett. 35, 85–90. (Correction: 50 (2000), 205.)Google Scholar
O'Hagan, A. and Leonard, T. (1976). Bayes estimation subject to uncertainty about parameter constraints. Biometrika 63, 201–203.Google Scholar
Pakes, A., Hung, W.-L. and Wu, J.-W. (2001). Criteria for the unique determination of probability distributions by moments. Austral. N. Z. J. Statist. 43, 101–111.Google Scholar
Schmoyeri, R. L., Beauchamp, J. J., Brandt, C. C. and Hoffman, F. O. Jr. (1996). Difficulties with the lognormal model in mean estimation and testing. Environm. Ecol. Statist. 3, 81–97.CrossRefGoogle Scholar
Slud, E. V. (1993). The moment problem for polynomial forms in normal random variables. Ann. Prob. 21, 2200–2214.Google Scholar
Stieltjes, T. J. (1894). Recherches sur les fractions continues. Ann. Fac. Sci. Univ. Toulouse 8, J1–J122 and 9 (1895), A5–A47. Reprinted in: Ann. Fac. Sci. Univ. Toulouse (6) 4 (1995), A5-A47.Google Scholar
Stoyanov, J. (1997). Counterexamples in Probability, 2nd edn. John Wiley, Chichester.Google Scholar
Stoyanov, J. (2000). Krein condition in probabilistic moment problems. Bernoulli 6, 939–949.Google Scholar
Stoyanov, J. (2004). Stieltjes classes for moment-indeterminate probability distributions. In Stochastic Methods and Their Applications (J. Appl. Prob. Spec. Vol. 41A), eds Gani, J. and Seneta, E., Applied Probability Trust, Sheffield, pp. 281–294.Google Scholar
Stoyanov, J. and Tolmatz, L. (2005). Method for constructing Stieltjes classes for M-indeterminate probability distributions. Appl. Math. Comput. 165, 669–685.Google Scholar
Williamson, M. and Gaston, K. J. (2005). The lognormal distribution is not an appropriate null hypothesis for the species-abundance distribution. J. Animal Ecology 74, 409–422.Google Scholar
Yamazaki, H. and Lueck, R. (1990). Why oceanic dissipation rates are not lognormal? J. Phys. Oceanography 20, 1907–1907.Google Scholar