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Biometrika 1999 86(4):871-880; doi:10.1093/biomet/86.4.871
© 1999 by Biometrika Trust
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On prediction intervals based on predictive likelihood or bootstrap methods

P HallA1,A2, L PengA1,A3 and N TajvidiA4

A1 Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia A2 E-mail: peter.hall@anu.edu.au A3 E-mail: liang.peng@maths.anu.edu.au A4 Department of Mathematics, Linköping University, 581 83 Linköping, Sweden E-mail: nataj@mai.liu.se

We argue that prediction intervals based on predictive likelihood do not correct for curvature with respect to the parameter value when they implicitly approximate an unknown probability density. Partly as a result of this difficulty, the order of coverage error associated with predictive intervals and predictive limits is equal to only the inverse of sample size. In this respect those methods do not improve on the simpler, 'naive' or 'estimative' approach. Moreover, in cases of practical importance the latter can be preferable, in terms of both the size and sign of coverage error. We show that bootstrap calibration of both a naive and predictive-likelihood approaches increases coverage accuracy of prediction intervals by an order of magnitude, and, in the case of naive intervals, preserves that method's numerical and analytical simplicity. Therefore, we argue, the bootstrap-calibrated naive approach is a particularly competitive alternative to more conventional, but more complex, techniques based on predictive likelihood.

Key Words: approximate predictive likelihood; Bayesian methods; bootstrap calibration; bootstrap iteration; coverage accuracy; double bootstrap; estimate predictive likelihood; Pareto distribution


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