[R-sig-eco] Are likelihood approaches frequentist?

[Rubén Roa-Ureta]

Dave Hewitt wrote:
> As Ben pointed out, the key difference between pure likelihood approaches and
> frequentist approaches is the addition of a layer of "significance"
> assessment based on the idea of repeated experimentation. (The term
> "frequentist" has been stretched in a variety of directions now, perhaps due
> to lazy writing, so sometimes it is unclear what's included under the
> umbrella.)
>
>   
I think Donald Rubin gave the right term: sampling-distribution 
inference, because it is an inference based on inspection of the sample 
space. Frequentist is not precise because a likelihoodist can subscribe 
to a strictly frequentist view of probabilities (e.g. Edwards) but still 
think that probabilities are not the correct tool for inferential 
statements.
> In his 2001 book "In All Likelihood: Statistical Modelling and Inference
> Using Likelihood", Yudi Pawitan refers to pure likelihood inference as
> "Fisher's third way", a compromise between frequentist and Bayesian
> approaches that began with Fisher himself. Inference based strictly on the
> likelihood function is not probabilistic, so would not conform to either of
> these two other paradigms.
>   
It seems to me that in the area of inference, Fisher had three 
offspring: significance tests/confidence intervals, direct-likelihood 
and fiducial inference. W.r.t. the first child he was a bit embarrassed. 
He wrote in his 1959 book "Objection has sometimes been made that the 
method of calculating Confidence Limits by setting an assigned value 
such as 1% on the frequency of observing [the test statistic] or less  
[...] is unrealistic in treating the values less than [the test 
statistic] which have not been observed , in exactly the same manner as 
the value of the [the test statistic] which is the one that has been 
observed. This feature is indeed not very defensible, save as an 
approximation" (p. 68). His favorite child appeared to be fiducial 
inference, but not many people understood this. It looks like his 
favorite was ignored, while the one he was a bit embarrassed about 
prospered. But we have to see what happens with the other child, 
direct-likelihood, maybe it prevails at the end of the day.
[snip]
Rubén