| In multi-level modeling, occasionally improper or unreasonable parameter estimates are obtained using the usual regression or ordinary multi-level modeling techniques. Motivated by this estimation problem, this dissertation research develops and extends the Bayesian Hierarchical Modeling approach to produce shrinkage in the posterior parameter estimators, and thus improve the parameter estimation. Lindley and Smith [1972] described two types of regression models based on exchangeability assumptions on parameters for linear models in their classic paper: exchangeability between regressions and exchangeability within multiple regressions. By combining the two types of assumptions, the posterior estimators will have "Dual-shrinkage" properties (Shrinkage in two directions) and thus behave more properly. Markov Chain Monte Carlo (MCMC) sampling procedures are utilized to simulate the parameter posterior distributions. Specifically, Gibbs Sampling and Metropolis-Hastings within Gibbs Sampling algorithms are programmed in R to obtain the posterior estimates. Then the combined model is generalized to allow between and within correlation assumptions. Lastly, the model and estimation procedures are applied to a consumer packaged goods product data set. |
