Why we care:
What is important is that treating a random effect as
a fixed one can trick you into believing the evidence
is way stronger than really it is.
Imagine 10 people with 10000 blood pressure measurements
on each of their two arms. It looks like 200,000 observations.
You might conclude with high confidence that left and right
arms have different blood pressures. But really there are
only 10 people. If you had infinitely many measurements
on them you still would have uncertainty about how the results apply.
Intuitively this is obvious. We hope our statistical methods
take account of this. But a black box might not take proper
account of the difference between 10 subjects with 10000 observations
each and 10000 subjects with 10 observations each. Those are
very different.
For ANOVA like problems we know how to handle this with some precision.
We'd hate to get it wrong on some other problem.
Why we care:
The bootstrap goes badly wrong for 'crossed random effect' type data
where both rows and columns are like the subjects in the analysis
above.
We might find ourselves with a seriously wrong answer.
Also: we'd hate to have to take all the theory of
restricted maximum likelihood estimators that gets used
in the ANOVA case, and try to generalize it to other settings.
It's bad enough when everything is Gaussian and mildly unbalanced.
It would be awful for other models.
So we'd like to do something easy like the bootstrap.
Unfortunately
McCullagh
proves the non-existence of an exactly correct bootstrap
even for a sample mean over all cases.
McCullagh's paper is not easy to read. (How could it be
easy to prove that no conceivable resampling will work?)
The bootstrap described in the lecture notes is approximately
right though.
Here are some key papers on matrix completion in roughly chronological order:
Here are some references on discrete variables with many levels, particularly Zipf's law.
This spot is for articles that don't yet fit into a big enough category.