Due Mon Nov 11, 2013.
The data set
LakeHuron included in R
annual measurements of the level, in feet, of Lake Huron 1875-1972.
The name of the time series is
LakeHuron, for example,
does a time series plot.
The average water level over this period was
> mean(LakeHuron)  579.0041
Obtain a standard error for this estimate
mean(LakeHuron)) using the subsampling bootstrap
with subsample length b = 10. Assume that the sample mean
obeys the square root law (that is, the
square root of n),
and assume the time series is stationary.
There seems no reason other than historical to use the
lqsoptions. LMS estimation is of low efficiency (converging at rate n− 1 ⁄ 3) whereas LTS has the same asymptotic efficiency as an M estimator with trimming at the quartiles (Marazzi, 1993, p. 201). LQS and LTS have the same maximal breakdown value of
(floor((n-p)/2) + 1)/nattained if
floor((n+p)/2) <= quantile <= floor((n+p+1)/2). The only drawback mentioned of LTS is greater computation, as a sort was thought to be required (Marazzi, 1993, p. 201) but this is not true as a partial sort can be used (and is used in this implementation).
Thus it seems that LMS regression is the
Wrong Thing (with a capital W and a capital T), something that survives
only for historico-sociological reasons: it was invented first and most
people that have heard of robust regression at all have only heard of it.
To be fair to Efron and Tibshirani, the literature cited in the documentation
lmsreg is the same vintage as their book. So maybe they
thought they were using the latest and greatest.
Anyway, the problem is to redo the LMS examples using LTS.
What changes? Does LTS really work better here than LMS? Describe the differences you see and why you think these differences indicate LTS is better (or worse, if that is what you think) than LMS.
contains a vector
data from a heavy tailed distribution such that the
sample mean has
rate of convergence n1 ⁄ 3,
n^(1 / 3) * (theta.hat - theta)
has nontrivial asymptotics (
nontrivial here meaning it doesn't
converge to zero in probability and also is bounded in probability,
so n1 ⁄ 3 is the right rate) where
theta.hat is the sample mean and
is the true unknown population mean.
When the sample mean behaves as badly as this, the sample variance
behaves even worse (it converges in probability to infinity), but
robust measures of scale make sense, for example, the interquartile
range (calculated by the
IQR function in R, note that
the capital letters are not a mistake).
- Using subsample size b = 20, do a subsampling bootstrap
to estimate the distribution of the sample mean.
- Make a histogram of
theta.starmarking the point
- Make a histogram of
b^(1 / 3) * (theta.star - theta.hat)which is the analog in the
bootstrap worldof the distribution of the quantity having nontrivial asymptotics displayed above.
- Calculate the subsampling bootstrap estimate of the IQR
theta.hat, rescaling by the ratio of rates in the appropriate fashion.
in the back of the book are here.