Simulation of Bayesian Posterior Distributions of Parameters of Constrained Models

Constrained Maximum Likelihood (CML) is a new software module developed at Aptech Systems for the generation of maximum likelihood estimates of statistical models with general constraints on parameters. These constraints can be linear or nonlinear, equality or inequality. The software uses the Sequential Quadratic Programming method with various descent algorithms to iterate from a given starting point to the maximum likelihood estimates. Standard asymptotic theory asserts that statistical inference regarding inequality constrained parameters does not require special techniques because for a large enough sample there will always be a con dence region at the selected level of con dence that avoids the constraint boundaries. Su ciently large, however, can be quite large, in the millions of cases when the true parameter values are very close to these boundaries. In practice, our nite samples may not be large enough for con dence regions to avoid constraint boundaries, and this has implications for all parameters in models with inequality constraints, even for those that are not themselves constrained. The usual method for statistical inference, comprising the calculation of the covariance matrix of the parameters and constructing t-statistics from the standard errors of the parameters, fails in the context of inequality constrained parameters because con dence regions will not generally be symmetric about the estimates. When the con dence region impinges on the constraint boundary, it becomes truncated, possibly in a way that a ects the con dence limit. It is therefore necessary to compute con dence intervals rather than t-statistics. Previous work (R.J. Schoenberg, "Constrained Maximum Likelihood", Computational Economics, 1997) shows that con dence intervals computed by inversion of the likelihood ratio statistics (i.e., pro le likeihood con dence limits) fail when there are constrained nuisance parameters in the model. This paper describes the weighted likelihood bootstrap method of Newton and Raftery ("Approximate Bayesian inference with the weighted likelihood bootstrap", J.R. Statist. Soc. B, 56:3-48,1994). This method generates simulations of the Bayesian posterior of the parameters. Con dence limits produced from these simulations may be interpreted as Bayesian con dence limits.