Effects of the reference set on frequentist inferences
1 Division of Biostatistics, Department of Public Health & Preventive Medicine, Oregon Health & Sciences University, Mail Code CR145, Oregon 97239-3098, U.S.A. pierced{at}ohsu.edu, 2 Department of Statistics, University of Udine, via Treppo 18, I-33100 Udine, Italy. bellio{at}dss.uniud.it
We employ second-order likelihood asymptotics to investigate how ideal frequentist inferences depend on the probability model for the data through more than the likelihood function, referring to this as the effect of the reference set. There are two aspects of higherorder corrections to first-order likelihood methods, namely (i) that involving effects of fitting nuisance parameters and leading to the modified profile likelihood, and (ii) another part pertaining to limitation in adjusted information. Generally, each of these involves a first-order adjustment depending on the reference set. However, we show that, for some important settings, likelihood-irrelevant model specifications have a second-order effect on both of these adjustments; this result includes specification of the censoring model for survival data. On the other hand, for sequential experiments the likelihood-irrelevant specification of the stopping rule has a second-order effect on adjustment (i) but a firstorder effect on adjustment (ii). These matters raise the issue of what are ‘ideal’ frequentist inferences, since consideration of ‘exact’ frequentist inferences will not suffice. We indicate that to second order ideal frequentist inferences may be based on the distribution of the ordinary likelihood ratio statistic, without commonly considered adjustments thereto.
Key Words: Censoring model; Higher-order asymptotics; Likelihood asymptotics; Likelihood principle; Modified directed deviance; Modified profile likelihood; Sequential experiment.
Received June 2004. Revised November 2005.