Ronald Schoenberg
Aptech Systems, Inc.
and
The University of Washington
11 November 1996
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.
CML computes two classes of confidence intervals, by inversion of the Wald and likelihood ratio statistics, and by simulation of the distributions of the parameters by weighted and unweighted bootstrap. Confidence intervals computed by inverting the likelihood ratio statistic are also called profile likelihood confidence intervals.
While the inversion methods of computing confidence intervals produce regions of the correct shape, they do not take into account the affect the truncation has on the size, i.e., on the extent to which the observed proportion of the regions which fail to include the true value of the parameter matches the nominal level of confidence. The correction for the effects of the truncated regions on the size takes the form of an adjustment to the statistic used in the inversion. Thus, in circumstances to be described, the statistics used in the inversion of the Wald or likelihood ratio statistics are modified when the confidence region is within a specified distance of a constraint boundary.
For the construction of a confidence limits for a parameter, the following three cases must be considered: (1) when the parameter has a constraint placed on it and the limit is within a specified distance of the constraint boundary, (2) when the confidence region of a nuisance parameter correlated with the parameter which is on or near its constraint boundary, and (3) when the confidence regions of both the parameter of interest and the nuisance parameter are near constraint boundaries.
A method is proposed for solving the first size problem. Monte Carlo evidence is presented which shows that the solution for (1) is satisfactory for computing confidence limits by inversion of likelihood ratio and Wald statistics. There is no known solution for (2) and (3).
KEY WORDS: Maximum Likelihood, Inequality Constraints, Profile Likelihood Statistical Inference