Summary
The characteristics of the one-dimensional generalized hyperbolic distributions are discussed, and the questions of maximum likelihood estimation for the hyperbolic distribution are considered in some detail. Various ways of approximating a theoretical distribution by one of the hyperbolic or generalized hyperbolic distributions are outlined and as an application of this an approximation is obtained to the distribution of the sum of a sample of observations from the hyperbolic distribution.
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References
Abramowitz, M. and Stegun, I. A. (1965). Handbook of Mathematical Functions. Dover, New York.
Barndorff-Nielsen, O. (1978a). Hyperbolic distributions and distribution on hyperbolae. Scandinavian Journal of Statistics, 5, 151–157.
Barndorff-Nielsen, O. (1978b). Information and Exponential Families. Wiley, Chichester.
Barndorff-Nielsen, O. (1979). Hyperbolic likelihood. Research Report No. 45, Department of Theoretical Statistics, Aarhus University (to appear in Festschrift to C. R. Rao ).
Barndorff-Nielsen, O. and Blaesild, P. (1980). Hyperbolic distributions. Encyclopedia of Statistical Sciences. Wiley, New York.
Barndorff-Nielsen, O., Blaesild, P. and Schou, G. (1979). Anote on skewness and kurtosis for the hyperbolic distributions. Research Report No. 53, Department of Theoretical Statistics, Aarhus University.
Blaesild, P. and Jensen, J. L. (1981). Multivariate distributions of hyperbolic type. In Statistical Distributions in Scientific Work, C. Taillie, G. P. Patil, and B. Baldessari, eds. Reidel, Dordrecht-Holland.
Burridge, J. (1980). A note on maximum likelihood estimation for regression models using grouped data. Journal of the Royal Statistical Society, Series B, 42.
Erdélyi, A., et al. (1954). Tables of Integral Transforms, Vol. I. McGraw-Hill, New York.
Feller, W. (1971). An Introduction to Probability Theory and its Applications, Vol. II (second ed.). Wiley, New York.
Gradshteyn, I. S. and Ryzhik, I. M. (1965). Tables of Integrals, Series, and Products. Academic Press, New York.
Grosswald, E. (1976). The Student t-distribution of any degree of freedom is infinitely divisible. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 36, 103–109.
Ismail, M. E. H. (1977). Integral representations and complete monotonicity of various quotients of Bessel functions. Canadian Journal. Of Mathematics, 29, 1198–1. 207.
Jensen, J. L. (1980). On the hyperboloid distribution. Research Report No. 59, Department of Theoretical Statistics, Aarhus University.
Kendall, M. G. and Stuart, A. (1969). The Advanced Theory of Statistics, Vol. I (third edition). Griffin, London
Lorch, L. (1967). Inequalities for some Whittaker functions. Arch. Math. (Brno), 3, 1–9.
Michelson, A. A., Pease, F. G. and Pearson, F. (1935). Measurement of the velocity of light in a partial vacuum. Astrophys. J., 82, 26–61.
Romanowski, M. (1979). Random Errors in Observations and the Inference of Modulation on their Distribution. Verlag Konrad Wittwer, Stuttgart.
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© 1981 D. Reidel Publishing Company
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Barndorff-Nielsen, O., Blaesild, P. (1981). Hyperbolic Distributions and Ramifications: Contributions to Theory and Application. In: Taillie, C., Patil, G.P., Baldessari, B.A. (eds) Statistical Distributions in Scientific Work. NATO Advanced Study Institutes Series, vol 79. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-8549-0_2
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DOI: https://doi.org/10.1007/978-94-009-8549-0_2
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