Orthogonal Functions of Inverse Gaussian Distributions
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.
KeywordsVariance Function Laguerre Polynomial Inverse Gaussian Distribution Edgeworth Expansion Cumulant Generate Function
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- Kotz, S and Johnson, N.L. (1981), Encyclopedia of Statistical Science, New York, NY: Wiley.Google Scholar