## Extensions of Hierarchical Bayesian Shrinkage Estimation with Applications to a Marketing Science ProblemIn 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. |

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### Contents

List of Tables | 5 |

Hierarchical Bayes and Related Methods | 11 |

Analytical Development of the New Cases | 32 |

Parameter Estimation | 59 |

Applications and Analysis | 81 |

Conclusions and Future Research | 101 |

### Common terms and phrases

assumed Bayesian box plot bpr ppr dist Chapter coefficient estimates combined model conditional posterior convergence correlation covariates dist fnd dnf dnf fd tpr Empirical Bayes Estimate estimate 3j estimates are shrunk exchangeability assumption exchangeability between regressions Feature and Display fnd dnf fd Gibbs algorithm Gibbs Sampling Procedures given Harmonic Mean Hierarchical Bayes Estimator Hierarchical Bayes model Hierarchical Linear Model Hoerl and Kennard iterations jth group jth regression Least Squares OLS Lemma Lindley and Smith's linear regression Markov Chain MCMC MCMC Sample Metropolis-Hastings within Gibbs multiple regressions multivariate normally multivariate normally distributed number of observations obtained OLS estimates parameter estimates percentage of improper pet of bpri Pet of Improper Pooled Estimator posterior distribution posterior estimates posterior mean ppr dist fnd ppri predictor recent values regression model ridge estimator ridge parameter Ridge Regression Ridge Trace Plot shrinkage effects shrinkage estimators third stage prior vague prior variance vector WinBUGS Wishart distribution