It is well known that the distributions of the Wald and likelihood ratio
statistics are modified when the true
value of a constrained parameter being estimated is on a
constraint boundary (Gourieroux, et al., 1982, Self and Liang, 1987,
Wolak, 1991).
In finite samples these effects occur in the *region* of the
constraint boundary, specifically when the true value is
within of the constraint
boundary. This has consequences for the calculation of
the confidence limits described in the previous sections.

We are concerned here with the unidimensional problem; the determining the confidence limits of one parameter in the model, leaving all other parameters as ``nuisance'' parameters. This problem can be divided into three cases,

- parameter constrained, no nuisance parameters constrained,
- parameter unconstrained, one or more nuisance parameters constrained,
- parameter constrained, one or more nuisance parameters constrained.

For case 2, the statistics are distributed as weighted mixtures of chi-squares when the correlation of the constrained nuisance parameter with the unconstrained parameter of interest is greater than about .8. A correction for these effects is feasible. However, for finite samples, the effects on the statistics due to a true value of a constrained nuisance parameter being within of the boundary are greater and more complicated than the effects of actually being on the constraint boundary. There is no systematic strategy available for correcting for these effects.

For case 3, the references disagree. Gourieroux, et al. (1982) and Wolak (1991) state that the statistics are distributed as a mixture of chi-squares. However, Self and Liang (1987) show that when the distributions of the parameter of interest and the nuisance parameter are correlated, the distributions of the statistics are not chi-square mixtures.

- Case 1: Confidence Limits of Constrained Parameter of Interest
- Correction for Effects of True Value in Region of Constraint Boundary

Fri Sep 12 09:47:41 PDT 1997