The asymptotic efficiency of improved prediction intervals

We consider the Barndorff-Nielsen and Cox (1994, p. 319) method of modifying an estimative prediction interval to obtain an improved prediction interval with better conditional coverage properties. The parameter estimator, on which this improved interval is based, is assumed to have the same asymptotic distribution as the conditional maximum likelihood estimator. This improved interval depends strongly on the asymptotic conditional bias of this estimator, which can be very sensitive to small changes in this estimator. We show, however, that the asymptotic efficiency of this improved prediction interval does not depend on this bias.