Bayes estimation in the case of the inverse Gaussian distribution
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, 11-15 (1988; Zbl 0744.62038)]. In the present paper, the Bayes estimates for unknown \(\mu, \lambda\), for some functions of them, and for the density of the inverse Gaussian distribution and some functionals of this density are constructed.
scientific article; zbMATH DE number 1213179