Abstract
The paper considers numerical issues of the empirical Bayesian approach model, applied to estimation of small rates. The condition for non-singularity of Bayesian estimates is given and the convenient iterative algorithm for estimation is described. The clustering algorithm is also developed, using the property of Poisson-Gaussian model to treat probabilities of events in populations being the same, if the variance of probabilities is small. The approach considered is illustrated by an application to the analysis of homicides and suicides data in Lithuania, 2003–2004.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramovich M, Stegun IA (1968) Handbook of Mathematical Functions. Dover, New York
Bradley PC, Thomas AL, Bradley C (2000) Bayes and Empirical Bayes Methods for Data Analysis. Chapman & Hall/CRC
Clayton D, Kaldor J (1987) Empirical Bayes estimates of age-standardized relative risks for use in disease mapping. Biometrics 43(3):671–81
Dennis JE, Schnabel RB (1996) Numerical Methods for unconstrained optimization and nonlinear Equations. In Classics for Applied Mathematics, 16, SIAM
Kantorovich LV, Akilov GP (1982) Functional Analysis. Pergamon Press, Oxford
Knorr-Held L, Rasser G (1999) Bayesian detection of clusters and discontinuities in disease maping. Sonderforschungsbereich 386, Discussion Paper 107
Leite JG, Rodrigues J, Milan LA (2000) A Bayesian analysis for estimating the number of species in a population using nonhomogeneous Poisson process. Stat Probab Lett 48:153–161. doi:10.1016/S0167-7152(99)00198-4
Meza JL (2003) Empirical Bayes estimation smoothing of relative risks in disease mapping. J Statist Plann Inference 112:43–62. doi:10.1016/S0378-3758(02)00322-1
Quigley J, Bedford T, Walls L (2007) Estimating rate of occurrence of rare events with Empirical Bayes: A railway application. Reliab Eng Syst Saf 92:619–627. doi:10.1016/j.ress.2006.02.007
Sakalauskas L (1995) On Bayes analysis of small rates in medicine. Proc. of the Intern. Conf. “Computer Data Analysis and Modeling”, 1995, September 14–19, Minsk, vol. 1, pp. 127–130
Tsutakava RK, Shoop GL, Marienfield CJ (1985) Empirical Bayes estimation of cancer mortality rates. Stat Med 4:201–212. doi:10.1002/sim.4780040210
Yasui Y, Liu H, Benach J, Winget M (2000) An empirical evaluation of various priors in the empirical Bayes estimation of small area disease risks. Stat Med 19:2409–2420. doi:10.1002/1097-0258(20000915/30)19:17/18<2409::AID-SIM578>3.0.CO;2-U
Vaurioa LK, Jankala KE (2006) Evaluation and comparison of estimation methods for failuke rates and probabilities. Reliab Eng Syst Saf 91:209–221. doi:10.1016/j.ress.2005.01.001
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sakalauskas, L. On the Empirical Bayesian Approach for the Poisson-Gaussian Model. Methodol Comput Appl Probab 12, 247–259 (2010). https://doi.org/10.1007/s11009-009-9146-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-009-9146-2