Abstract
The univariate natural exponential families with quadratic variance functions have the orthogonal polynomial systems which are generated by differentiating the densities. This method, however, is not applicable to inverse Gaussian distributions because their variance functions are cubic. We will generate non-orthogonal but simple polynomials and orthogonal functions of inverse Gaussian distributions based on Laguerre polynomials. Properties of the polynomials and the functions are obtained by the use of the generating functions. They are applied to approximate a lognormal density and examined numerically.
Keywords
- Variance Function
- Laguerre Polynomial
- Inverse Gaussian Distribution
- Edgeworth Expansion
- Cumulant Generate Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Barndorff-Nielsen, O.E. (1978), Information and Exponential Families in Statistical Theory, New York, NY: Wiley.
Chhikara, R.S. and Folks, J.L. (1989), The Inverse Gaussian Distribution, New York, NY: Marcel Dekker, INC.
Crow, E.L. and Shimizu, K. (1988), Lognormal Distributions, New York, NY: Marcel Dekker, INC.
Kotz, S and Johnson, N.L. (1981), Encyclopedia of Statistical Science, New York, NY: Wiley.
Letac, G. and Mora, M. (1990), “Natural real exponential families with cubic variance functions”, Ann. Statist., 18, 1–37.
Morris, C.N. (1982), “Natural exponential families with quadratic variance functions”, Ann. Statist., 10, 65–80.
Nishii, R. (1993), “Convergence of the Gram-Charlier expansion after the normalizing Box-Cox transformation”, Ann. Inst. Statist. Math., 45, 173–186.
Whitmore, G.A. and Yalovsky, M. (1978), “A normalizing logarithmic transformation for inverse Gaussian random variables”, Technornetrics, 20, 207–208.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Nishii, R. (1996). Orthogonal Functions of Inverse Gaussian Distributions. In: Jewell, N.P., Kimber, A.C., Lee, ML.T., Whitmore, G.A. (eds) Lifetime Data: Models in Reliability and Survival Analysis. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-5654-8_32
Download citation
DOI: https://doi.org/10.1007/978-1-4757-5654-8_32
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-4753-6
Online ISBN: 978-1-4757-5654-8
eBook Packages: Springer Book Archive