MULTIMODALITY p**-FORMULA AND CONFIDENCE REGIONS

Abstract

Barndorff-Nielsen’s celebrated p*-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 a crucial adjustment to the basic formula. The adjustment is based on a simple idea and straightforward to implement, yet delivers important improvements. It is based on recognizing that certain outcomes are theoretically impossible and the density of the MLE should then equal zero, rather than the positive density that a straight application of p* would suggest. This has implications for inference and we show how to use the new p**-formula to construct improved confidence regions. These can be disjoint as a consequence of the bimodality. The degree of bimodality depends heavily on the value of an approximate ancillary statistic and conditioning on the observed value of this statistic is therefore desirable. The p**-formula naturally delivers the relevant conditional distribution. We illustrate these results in small and large samples using a simple nonlinear regression model and errors in variables model where the measurement errors in dependent and explanatory variables are correlated and allow for weak proxies.