In simple linear regression, it is assumed that two variables are linearly related, with unknown intercept and slope parameters. In particular, a regressor variable is assumed to be precisely measurable, and a response is assumed to be a random variable whose mean depends on the regressor via a linear function. For the simple linear regression problem, interest typically centers on estimation of the unknown model parameters, and perhaps application of the resulting estimated linear relationship to make predictions about future response values corresponding to given regressor values. The linear statistical calibration problem (or, more precisely, the absolute linear calibration problem), bears a resemblance to simple linear regression. It is still assumed that the two variables are linearly related, with unknown intercept and slope parameters. However, in calibration, interest centers on estimating an unknown value of the regressor, corresponding to an observed value of the response variable. We consider Bayesian methods of analysis for the linear statistical calibration problem, based on noninformative priors. Posterior analyses are assessed and compared with classical inference procedures. It is shown that noninformative prior Bayesian analysis is a strong competitor, yielding posterior inferences that can, in many cases, be correctly interpreted in a frequentist context. We also consider extensions of the linear statistical calibration problem to polynomial models and multivariate regression models. For these models, noninformative priors are developed, and posterior inferences are derived. The results are illustrated with analyses of published data sets. In addition, a certain type of heteroscedasticity is considered, which relaxes the traditional assumptions made in the analysis of a statistical calibration problem. It is shown that the resulting analysis can yield more reliable results than an analysis of the homoscedastic model. / Ph. D.

Bayesian reference analysis is a method of determining the prior under the Bayesian paradigm. It incorporates as little information as possible from the experiment. Estimation of the ratio of two independent Poisson rates is a common practical problem. In this thesis, the method of reference analysis is applied to derive the posterior distribution of the ratio of two independent Poisson rates, and then to construct point and interval estimates based on the reference posterior. In addition, the Frequentist coverage property of HPD intervals is verified through simulation.