Sampling-based structural inference on linear transformation models with non-normal errors via approximate Bayesian computation

Abstract

Structural inference back in the late 1960’s, can be applied to general linear models with non-normal errors by separating the unknown error parameter (intrinsic parameter) from the regression and scale parameters (transformation parameters) due to their different natures. A natural analysis of the underlying error vector, randomness source in the model, by a change of coordinates for the model, similar to a change of reference system for relativity theory in physics, decomposes the error vector into an observable component of (n−p−1) dimension and an unobservable component of (p+1) dimension. This leads to a legitimate marginal likelihood function for the intrinsic parameter based on the observed component of the error vector and the corresponding conditional distribution of the unobservable component given the observed for inferences of the transformation parameters.

The popular sampling technique of the Approximate Bayesian Computation methodology (ABC) in Bayesian regime is employed in this thesis to deal with two situations. When the error density does not have any parameter, the structural model is named as the transformation model. We study the inferential sampling from the conditional distribution, which is known up to the normalizing constant. Several asymptotic results of the sampling conditional density regarding the filter size and the sample size are proved via the equivalent spherical coordinate transformation. We demonstrate the application with both real and simulated data sets and compare inferences with widely-used frequentist methods.

When the error density has an unknown parameter, called intrinsic parameter, or simply error parameter, we introduce a mixture of Bayesian and structural inference by introducing a prior of the intrinsic parameter for Bayesian inference, then followed by non-Bayesian conditioning inference for the transformation parameters, and using ABC for sampling from both inferential distributions. Several asymptotic results of the sampling posterior with respect to the filter size and the number of sampled observations are proved via the equivalent spherical coordinate transformation as well. We apply this mixture approach for inference to real data sets and compare it with other widely used methods by simulation study.