The Annals of Mathematical Statistics

Statistical Properties of Inverse Gaussian Distributions. I

M. C. K. Tweedie

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Abstract

A report is presented on some statistical properties of the family of probability density functions $$\exp \lbrack -\lambda(x - \mu)^2/2\mu^2x\rbrack\lbrack\lambda/2\pi x^3\rbrack^{1/2}$$ for a variate $x$ and parameters $\mu$ and $\lambda$, with $x, \mu, \lambda$ each confined to $(0, \infty)$. The expectation of $x$ is $\mu$, while $\lambda$ is a measure of relative precision. The chief result is that the ml estimators of $\mu$ and $\lambda$ have stochastically independent distributions, and are of a nature which permits of the construction of an analogue of the analysis of variance for nested classifications. The ml estimator of $\mu$ is the sample mean, and for a fixed sample size $n$ its distribution is of the same family as $x$, with the same $\mu$ but with $\lambda$ replaced by $\lambda n$. The distribution of the ml estimator of the reciprocal of $\lambda$ is of the chi-square type. The probability distribution of $1/x$, and the estimation of certain functions of the parameters in heterogeneous data, are also considered.

Article information

Source
Ann. Math. Statist., Volume 28, Number 2 (1957), 362-377.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177706964

Digital Object Identifier
doi:10.1214/aoms/1177706964

Mathematical Reviews number (MathSciNet)
MR110132

Zentralblatt MATH identifier
0086.35202

JSTOR
links.jstor.org

Citation

Tweedie, M. C. K. Statistical Properties of Inverse Gaussian Distributions. I. Ann. Math. Statist. 28 (1957), no. 2, 362--377. doi:10.1214/aoms/1177706964. https://projecteuclid.org/euclid.aoms/1177706964


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See also

  • Part II: M. C. K. Tweedie. Statistical Properties of Inverse Gaussian Distributions. II. Ann. Math. Statist., Volume 28, Number 3 (1957), 696--705.