Weighted least squares regression istied to normality of the residual distribution. The two main
motivations for weights are scale families and convolution
families. In normal-theory regression, normality is
retained under both scaling and convolution. Outside of
normal-theory regression, the two motivations lead to
different models and inferences. Under a scale model, the shape of the
residual distribution remains the same; under a
convolution model, the distribution moves toward normality
as more units are convolved. Empirically, we have observed
both the scale family and the convolution family. Some data sets
show slower movement toward normality than convolution
would suggest.
In this thesis, we develop a family of semi-parametric
Bayesian models that include the scale and convolution
families as extreme points. Mean and variance constraints are placed on the residual distribution. The constrained aggregation models rely on the centered stick-breaking process. A novel computational strategy is presented in which proposal distributions are constructed to reflect features of the posterior distribution and improve mixing of the Markov chain Monte Carlo algorithm. The acceptance probabilities of Metropolis-Hastings steps are carefully calculated to take into account the constraints imposed on the model. Two simulation studies and fits to real data sets show that the constrained semi-parametric Bayesian models have good predictive performance.