Ronald Schoenberg
Aptech Systems, Inc.
and
The University of Washington
August, 1997
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 confidence region at the selected level of confidence that avoids the constraint boundaries. Sufficiently 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 finite samples may not be large enough for confidence 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 confidence regions will not generally be symmetric about the estimates. When the confidence region impinges on the constraint boundary, it becomes truncated, possibly in a way that affects the confidence limit. It is therefore necessary to compute confidence intervals rather than t-statistics.
Previous work (R.J. Schoenberg, "Constrained Maximum Likelihood", Computational Economics, 1997) shows that confidence intervals computed by inversion of the likelihood ratio statistics (i.e., profile likeihood confidence 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. Confidence limits produced from these simulations may be interpreted as Bayesian confidence limits.
KEY WORDS: Maximum Likelihood, Inequality Constraints, Bayesian Statistical Inference