This paper studies the least upper bounds on coverage probabilities of the empirical likelihood ratio confidence regions based on estimating equations. The implications of the bounds on empirical likelihood inference are also discussed.
A Bayesian analysis of the evidence for human-induced climate change in global surface temperature observations is described. The analysis uses the standard optimal detection approach and explicitly incorporates prior knowledge about uncertainty and the influence of humans on the climate. This knowledge is expressed through prior distributions that are noncommittal on the climate change question. Evidence for detection and attribution is assessed probabilistically using clearly defined criteria. Detection requires that there is high likelihood that a given climate-model-simulated response to historical changes in greenhouse gas concentration and sulphate aerosol loading has been identified in observations. Attribution entails a more complex process that involves both the elimination of other plausible explanations of change and an assessment of the likelihood that the climate-model-simulated response to historical forcing changes is correct. The Bayesian formalism used in this study deals with this latter aspect of attribution in a more satisfactory way than the standard attribution consistency test. Very strong evidence is found to support the detection of an anthropogenic influence on the climate of the twentieth century. However, the evidence from the Bayesian attribution assessment is not as strong, possibly due to the limited length of the available observational record or sources of external forcing on the climate system that have not been accounted for in this study. It is estimated that strong evidence from a Bayesian attribution assessment using a relatively stringent attribution criterion may be available by 2020.
We extend the empirical likelihood of Owen [Ann. Statist. 18 (1990) 90-120]
by partitioning its domain into the collection of its contours and mapping the
contours through a continuous sequence of similarity transformations onto the
full parameter space. The resulting extended empirical likelihood is a natural
generalization of the original empirical likelihood to the full parameter
space; it has the same asymptotic properties and identically shaped contours as
the original empirical likelihood. It can also attain the second order accuracy
of the Bartlett corrected empirical likelihood of DiCiccio, Hall and Romano
[Ann. Statist. 19 (1991) 1053-1061]. A simple first order extended empirical
likelihood is found to be substantially more accurate than the original
empirical likelihood. It is also more accurate than available second order
empirical likelihood methods in most small sample situations and competitive in
accuracy in large sample situations. Importantly, in many one-dimensional
applications this first order extended empirical likelihood is accurate for
sample sizes as small as ten, making it a practical and reliable choice for
small sample empirical likelihood inference.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1143 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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