Conditional simulation for regression-scale models

A regression-scale model is of the form y = X\beta + \sigma\epsilon, where X is a fixed nxp design matrix, \beta\in\Real^p an unknown regression coefficient, \sigma>0 a scale parameter, and \epsilon represents an n-dimensional vector of errors whose density is known. Inference is usually made conditionally on the sample configuration a = (y - X\hat\beta)/\hat\sigma, where (\hat\beta, \hat\sigma) are the maximum likelihood estimates. Higher-order asymptotics provide very accurate approximations to exact conditional procedures thus avoiding multidimensional numerical integration. This paper presents how by means of the Metropolis-Hastings algorithm, a powerful Markov chain Monte Carlo technique which allows to simulate from distributions that are only known up to the normalizing constant, conditional properties of these methods can be assessed.