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BC_{a} stands for bias corrected and accelerated
.
It is an example of really horrible alphabet soup
terminology.
Really trendy, though. Used to be that scientists used terminology
that involved real English (or Latin) words. Nowadays, it is trendy
to just use letters. It's molecular biology envy (a la
DNA, RNA, G6PD, and so forth). If you can actually express yourself
and be understood, then you must not be a real scientist, because as
everyone knows science is hard to understand. Hence the modern trend
for scientists to speak and write as illiterately as possible.
To parody this trend, we call these alphabet soup, type 1
intervals
(for type 2
see below).
library(bootstrap)
says we are going to
use code in the bootstrap
library, which is not available
without this command. Here library(bootstrap)
is necessary
for two reasons. Without it we can't get the data spatial
(on-line
help)
and we also can't get the function bcanon
(on-line
help) that we use to
construct BC_{a} intervals.
theta
is a function that calculates the
point estimate on which the interval is based. Here the point estimate
is the variance of the empirical distribution, calculated by the function
evar
.
nboot = 1000
is because of the notion that in general
we need a large bootstrap sample size for confidence intervals. Also we
have to supply this argument. There is no default.
These are the alphabet soup, type 2
intervals.
ABC stands for
approximate bootstrap confidence
, whatever that means. It doesn't
actually bootstrap, but just approximates the bootstrap. Chapter 22 of
Efron and Tibshirani explains, but we won't get into that.
The rather strange form of rvar
is an estimator written in resampling form
, which we saw
before in the improved bootstrap bias correction
procedure.
As the example shows and the
on-line
help documents, the tt
argument to the abcnon
function must have the signature function(p, x)
where
x
is the data
p
is a probability vector the same length as the data.
The idea is that the relationship of a bootstrap sample x.star
to the original data x
can be expressed as a probability
vector p.star
such that p.star[i]
is the fraction
of times x[i]
occurs in x.star
.
We have to write a function that calculates the estimator given x
and p.star
.
And this function must work for any probability vector
p.star
, not just ones with elements that are multiples of
1 / n
, because that's what the ABC method requires.
Unfortunately, this is, in general, hard.
Fortunately, this is, for moments, quite straightforward.
For any function g
, any data vector x
,
and any probability vector p
, the expression
sum(g(x) * p)
calculates the expectation of the random variable g(X)
in the probability model that assigns probability p[i]
to the
point x[i]
for each i
(and probability zero to
everywhere else).
Thus
sum(x * p)
calculates the mean
sum((x - a)^2 * p)
calculates the second moment about the point a
, and so forth.