Explanation of answers to Quiz #7
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 and 2nd questions of the quiz have 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).
-
Question 1
deals with
using the square of the sample mean to estimate the
square of the distribution
mean.
-
Jackknifing
should be used since
the estimator is a smooth plug-in quadratic estimator.
-
Question 2
deals with
using the square of the sample mean to estimate the
2nd moment of the distribution.
-
Bootstrapping
should be used since
the estimator is a not the plug-in estimator.
The 3rd 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 3
deals with
using the sample 2nd moment to estimate the square of the mean
of the distribution.
-
Jackknifing
should be used since
the sample 2nd moment is a smooth linear estimator.
-
Question 4
deals with
using the inverse of the sample mean to estimate the inverse
of the distribution
mean.
-
Bootstrapping
should be used since
the inverse of the sample mean is not linear.
-
Question 5
deals with
using the sample correlation coefficient to estimate the distribution
correlation coefficient.
-
Bootstrapping
should be used since
the sample correlation coefficient is not linear.