Bayes estimation in the case of the inverse Gaussian distribution
type: Item
instance of: scholarly article
title: Bayes estimation in the case of the inverse Gaussian distribution
zbMATH Open document ID: 0906.62025
DOI: 10.1007/BF02365368
author: E. V. Mazanova
published in: Journal of Mathematical Sciences (New York)
publication date: 1998-11-24T00:00:00Z
review text: In statistical data processing, in the case of asymmetric nonnegative observations, a two-parameter inverse Gaussian distribution is widely used. It can be applied in some probabilistic models of real phenomena, in particular, in the problem of passage across a fixed level by the Wiener process. The density of the inverse Gaussian distribution is representable in the form \[ p(x | \mu,\lambda) ={\lambda^{1/2} \over (2\pi)^{1/2} x^{3/2}} \exp \left\{-{\lambda ( x-\mu)^2 \over 2\mu^2x} \right\}, \quad x>0,\;\mu>0,\;\lambda>0. \tag{1} \] It has been shown by \textit{J. L. Folks} and \textit{R. S. Chhikara} [J. R. Stat. Soc., Ser. B 40, 263-289 (1978; Zbl 0408.62011)] that the complete sufficient statistic for the family (1) are the following independent random variables: \[ \overline x_n= {1\over n} \sum^n_{i=1} x_i, \quad V_n= \sum^n_{i=1} \left[{1 \over x_i} -{1\over \overline x} \right], \] where the first random variable has the distribution (1) and the second one is distributed as a chi-square random variable with \((n-1)\) degrees of freedom \((V_n \sim \chi^2_{n-1} (1/ \lambda))\). It can be seen that \(\mu\) is the mean and \(\mu^3/ \lambda\) is the variance of the random variables having the inverse Gaussian distribution. Estimates for the unknown parameters \(\mu,\lambda\) and for some functions of them were found, e.g., by \textit{M. M. Larin} [Eng. Cybern. 20, No. 6, 65-68 (1982); translation from Izv. Akad. Nauk SSSR, Tekh. Kibern. 1982, No. 6, 134-137 (1982; Zbl 0524.62029)]. In particular, the estimates \(\mu_x= {\mathbf E} [X]\) and \(\sigma^2_x ={\mathbf D} [X]\) for \(\mu, \lambda\) were obtained in that paper in the case of the inverse Gaussian distribution. An unbiased estimate for the density (1) when both parameters are unknown and for some functionals of this density was obtained by \textit{A. P. Vedernikova} and \textit{Ya. P. Lumel'skij} [J. Sov. Math. 56, No. 3, 2407-2409 (1991); translation from Interuniv. Collect. Sci. Works, Perm'/USSR, 11-…
Mathematics Subject Classification ID: 62F15
Mathematics Subject Classification ID: 62F10
zbMATH DE Number: 1213179
MaRDI profile type: Publication
cites work: Q3036474
cites work: Uniformly Minimum Variance Unbiased Estimation for the Inverse Gaussian Distribution
cites work: Umvu estimators of the mode and limits of an interval for the inverse gaussian distribution
cites work: Q3965385
cites work: Q5580725
cites work: Q4195757
full work available at URL: https://doi.org/10.1007/bf02365368
OpenAlex ID: W2063129551
description: scientific article; zbMATH DE number 1213179