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## Problems with Confidence Limits using Inversion

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,

1. parameter constrained, no nuisance parameters constrained,
2. parameter unconstrained, one or more nuisance parameters constrained,
3. parameter constrained, one or more nuisance parameters constrained.
For case 1, when the true value is on the boundary, the statistics are distributed as a simple mixture of two chi-squares. Monte Carlo evidence presented below will show that this holds as well in finite samples for true values within of the constraint boundary.

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.   Next: Case 1: Confidence Limits Up: Confidence Limits by Inversion Previous: Confidence Limits by Inversion

R. Schoenberg
Fri Sep 12 09:47:41 PDT 1997