## Bayesian Analysis

### Bayesian Estimation of Log-Normal Means with Finite Quadratic Expected Loss

#### Abstract

The log-normal distribution is a popular model in biostatistics and other fields of statistics. Bayesian inference on the mean and median of the distribution is problematic because, for many popular choices of the prior for the variance (on the log-scale) parameter, the posterior distribution has no finite moments, leading to Bayes estimators with infinite expected loss for the most common choices of the loss function. We propose a generalized inverse Gaussian prior for the variance parameter, that leads to a log-generalized hyperbolic posterior, for which it is easy to calculate quantiles and moments, provided that they exist. We derive the constraints on the prior parameters that yield finite posterior moments of order $r$. We investigate the choice of prior parameters leading to Bayes estimators with optimal frequentist mean square error. For the estimation of the lognormal mean we show, using simulation, that the Bayes estimator under quadratic loss compares favorably in terms of frequentist mean square error to known estimators.

#### Article information

Source
Bayesian Anal., Volume 7, Number 4 (2012), 975-996.

Dates
First available in Project Euclid: 27 November 2012

https://projecteuclid.org/euclid.ba/1354024469

Digital Object Identifier
doi:10.1214/12-BA733

Mathematical Reviews number (MathSciNet)
MR3000021

Zentralblatt MATH identifier
1330.62121

#### Citation

Fabrizi, Enrico; Trivisano, Carlo. Bayesian Estimation of Log-Normal Means with Finite Quadratic Expected Loss. Bayesian Anal. 7 (2012), no. 4, 975--996. doi:10.1214/12-BA733. https://projecteuclid.org/euclid.ba/1354024469

#### References

• Abramowitz, A. and Stegun, I. (1968). Handbook of mathematical functions. Dover, NY: Addison–Wesley, 5th edition edition.
• Barndorff-Nielsen, O. (1977). “Exponentially decreasing distributions for the logarithm of particle size.” Proceedings of the Royal Statistical Society, A353: 401–419.
• Bibby, B. and M., S. (2003). “Generalized hyperbolic and inverse Gaussian distributions: limiting cases and approximation of processes.” In Rachev, S. (ed.), Handbook of Heavy Tailed Distributions in Finance, 211–248. New York, NY: Elsevier Science B.V.
• Breymann, W. and Lüthi, D. (2010). ghyp: a package on Generalized Hyperbolic distributions. availalble at cran.r-project.org/web/packages/ghyp/index.html.
• Dagne, G. (2001). “Bayesian tranformed models for small area estimation.” TEST, 10: 375–391.
• Eberlein, E. and von Hammerstein, E. (2004). “Hyperbolic processes in finance.” In Dalang, R., Dozzi, M., and F., R. (eds.), Seminar on Stochastic Analysis, Random Fields and Applications IV, Progress in Probability, 221–264. Berlin, DE: Birkhäuser Verlag.
• EPA (2002). Limits for exposure point concentrations at hazardous waste sites. Washington, DC: OSWER 9285.6-10.
• Evans, I. and Shaban, S. (1976). “New estimators (of smaller MSE) for parameters of a lognormal distribution.” Biometrische Zeitschrift, 18: 453–466.
• Gelman, A. (2006). “Prior distributions for variance parameters in hierarchical models.” Bayesian Analysis, 1: 515–533.
• Gill, P. (2004). “Small-sample inference for the comparison of means of log-normal distributions.” Biometrics, 60: 525–527.
• Krishnamoorthy, K., Mallick, A., and Mathew, T. (2011). “Inference for the Lognormal mean and quantiles based on samples with left and right type I censoring.” Technometrics, 51: 72–83.
• Laforgia, A. and Natalini, P. (2010). “Some inequalities for modified Bessel functions.” Journal of Inequalities and Applications, Article ID 253035: 10pages.
• Limpert, E., Stahel, W., and Abbt, M. (2001). “Log-normal distributions across the sciences: keys and clues.” BioScience, 51: 341–352.
• Nguyen, T., Chen, J., Gupta, A., and Dinh, K. (2003). “A proof of the conjecture on positive skeweness of Generalized Inverse Gaussian distributions.” Biometrika, 90: 245–250.
• Rukhin, A. (1986). “Improved estimation in lognormal models.” Journal of the American Statistical Association, 81: 1046–1049.
• Shen, H., Brown, L., and Zhi, H. (2006). “Efficient estimation of log-normal means with application to pharmacokinetic data.” Statistics in Medicine, 25: 3023–3038.
• Singh, A., Singh, A., and Engelhardt, M. (2002). The lognormal distribution in environmental applications. Washington, DC: EPA/600/R-97/006.
• Zellner, A. (1971). “Bayesian and non-Bayesian analysis of the log-normal distribution and log-normal regression.” Journal of the American Statistical Association, 66: 327–330.
• Zhou, X. and Gao, S. (1997). “Confidence intervals of the log-normal mean.” Statistics in Medicine, 16: 783–790.
• Zou, G., Taleban, J., and Huo, C. (2009). “Confidence interval estimation for lognormal data with application to health economics.” Computational Statistics and Data Analysis, 53: 3755–3764.