Explanation of answers to Quiz #5
You were supposed to choose the jackknife when both methods perform
about equally well, since the jackknife has the advantage of being
nonrandom and quicker (when n is smaller than what one should use
for B, and even for some larger values of n since
bootstrapping also makes extensive use of a random number generator
which adds to the time required (since with bootstrapping one has to
generate B bootstrap samples and compute B replicates,
whereas with jackknifing one just has to compute n replicate
values of the statistic)).
The 1st question of the quiz has to do with estimating bias. The
jackknife should be chosen for estimating the bias of an estimator if
the estimator is
- the plug-in estimator,
- and smooth (so not a sample median, or a sample trimmed mean),
- and quadratic (noting that
a linear plug-in estimator
is guaranteed to be unbiased for the estimand, and so no estimate of
bias should be needed for
a linear plug-in estimator).
- Specifically,
Question 1
deals with
using the sample mean to estimate the distribution
median.
-
Bootstrapping
should be used since
the sample mean is not the plug-in estimator.
The 2nd through the 5th
questions of the quiz have to do with estimating standard error. The
jackknife should be chosen for estimating the standard
error of an estimator if
the estimator is
- smooth (so not a sample median, or a sample trimmed mean),
- and linear (like the sample mean, or the sample 2nd moment).
-
Question 2
deals with
using the sample median to estimate the distribution
median.
-
Bootstrapping
should be used since
the sample median is not smooth.
-
Question 3
deals with
using a sample 10% trimmed mean to estimate the distribution
mean.
-
Bootstrapping
should be used since
the sample 10% trimmed mean is not smooth.
-
Question 4
deals with
using the sample 2nd moment
to estimate the distribution
2nd moment.
-
Jackknifing
should be used since
the sample 2nd moment is both smooth and linear.
-
Question 5
deals with
using the sample variance to estimate the distribution
variance.
-
Bootstrapping
should be used since
the sample variance is not linear (it's quadratic).