Kees Jan van Garderen (University of Amsterdam)

 'Multimodality Adjusted p-star Formula and Confidence Regions'

Abstract:

Barndorff-Nielsen's celebrated p-star formula and variations thereof have amongst their various attractions the ability to approximate bimodal distributions. In this paper we show that in general this requires an important adjustment to the basic formula that is easy to implement. We partition the sample space and show that certain regions of the sample space imply zero density for the MLE, rather than the positive density that a straight application of p-star formula would suggest. The p-star formulas automatically give a conditional distribution and we illustrate the importance of conditioning by showing that the degree of bimodality in our examples depends heavily on the value of an approximate ancillary. Bimodality has consequences for inference and we use the p-star formula to obtain improved confidence regions for the parameters of economics interest. We illustrate these results with the well-known single structural equations model and the model by Hall and Horowitz (1996).