Effect of the Reference Set on Frequency Inference

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Effect of the Reference Set on Frequency Inference

• Donald A. Pierce
• Radiation Effects Research Foundation, Japan
• Ruggero Bellio
• Udine University, Italy

Paper, this talk, other things at
http//home.att.ne.jp/apple/pierce/
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Here we study to what extent, and in what manner,
second-order inferences depend on the reference
set
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Example Sequential Clinical Trials
The likelihood function does not depend on the
stopping rule, including that with fixed n .
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This involves what is called the censoring model.
First-order inferences depend on only the
likelihood function and not on the censoring
model.
In what way (how much) do higher-order
inferencesdepend on the censoring model? It is
unattractive that they should depend at all on
this.
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Typical second-order effects
Generally, in settings with substantial numbers
of nuisance parameters, and even for large
samples, adjustments may be much larger than this
--- or they may not be
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Some general notation and concepts
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It turns out that each of these approximations
has a leading term depending on only the
likelihood function, with a next term of one
order smaller depending on the reference set
A similar expansion gives the same result for the
other sample-space derivative
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Thus, in this independence case, second-order
inference, although not determined by the
likelihood function, is determined by the
contributions to it
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A main application of this pertains to censoring
models, if censoring and response times for
individuals are stochastically independent
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Things are quite different for comparing
sequential and fixed sample size experiments ---
usually cannot have contributions that are
independent in both reference sets
But first we need to consider under what
conditions second-order likelihood asymptotics
applies to sequential settings
We argue in our paper that it does whenever usual
first-order asymptotics applies
These conditions are given by Anscombes Theorem
A statistic asymptotically standard normal for
fixed n remains so when (a) the CV of n
approaches zero, and (b) the statistic is
asymptotically suitably continuous. Discrete n in
itself does not invalidate (b)
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Thus, in sequential experiments the NP adjustment
and MPL do not depend on the stopping rule, but
the INF adjustment does
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SUMMARY
When there are contributions to the likelihood
that are independent under each of two reference
sets, then second-order ideal frequency inference
is the same for these.
In sequential settings we need to consider the
nuisance parameter and information adjustments.
To second order, the former and the modified
profile likelihood do not depend on the stopping
rule, but the latter does.
This is all as one might hope, or expect.
Inference should not, for example, depend on the
censoring model but it should depend on the
stopping rule
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Appendix Basis for higher-order likelihood
asymptotics