Paper published: Bayesian Simultaneous Intervals for Small Areas: An Application to Variation in Maps

Bayesian Simultaneous Intervals for Small Areas: An Application to Variation in Maps Erik Barry Erhardt, Balgobin Nandram, Jai Won Choi International Journal of Statistics and Probability Vol 1, No 2, pp. 229–243 Received: September 19, 2012 Accepted: October 24, 2012 Online: October 29, 2012 doi:10.5539/ijsp.v1n2p229 Abstract Bayesian inference about small areas is of considerable current interest, and simultaneous intervals for the parameters for the areas are needed because these parameters are correlated. This is not usually pursued because with many areas the problem becomes difficult. We describe a method for finding simultaneous credible intervals for a relatively large number of parameters, each corresponding to a single area. Our method is model based, it uses a hierarchical Bayesian model, and it starts with either the 100(1-alpha)% (e.g., alpha=0.05 for 95%) credible interval or highest posterior density (HPD) interval for each area. As in the construction of the HPD interval, our method is the result of the solution of two simultaneous equations, an equation that accounts for the probability content, 100(1-alpha)% of all the intervals combined, and an equation that contains an optimality condition like the “equal ordinates” condition in the HPD interval. We compare our method with one based on a nonparametric method, which as expected under a parametric model, does not perform as well as ours, but is a good competitor. We illustrate our method and compare it with the nonparametric method using an example on disease mapping which utilizes a standard Poisson regression model.

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