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Stat 511
Homework 7 Solution
Spring 2011
Due: 11am, Monday Mar 7
1. Hazardous waste sites often contain multiple nasty chemicals. At one site in New Jersey,
the state regulators needed to assess the correlation between the concentration of hexavalent
Chromium, Cr
+6
, and Antimony, Sb. Evaluating this correlation is complicated by the pres
ence of
<
detection limit values for both chemicals. The data in bootCr.txt contain 10,000
bootstrap estimates of the correlation and the variance in the correlation. These are derived
using a maximumlikelihood estimator of the correlation and nonparametric bootstrap resam
pling. The estimated correlation is
r
=0
.
827 and the estimated variance of the correlation is
0.00517. You do not need to estimate the correlation from a data set; I have done that and
the bootstrap sampling for you.
(a) Calculate the percentile bootstrap 99% conFdence interval from these bootstrap samples.
(0.617,0.976)
(b) Calculate the percentilet bootstrap 99% conFdence interval.
(0.665,1.012)
(c) Which interval is the more appropriate to report?
Based on the plot below, there appears to be a negative correlation between std and the
estimate. The variance of the estimator is correlated with the estimate, so based on what
I said in lecture, the percentilet bootstrap 99% conFdence interval is more appropriate
to report even though it is not range respecting. This leads to an interval of (0.665,1).
0.5
0.6
0.7
0.8
0.9
1.0
0.05
0.06
0.07
0.08
0.09
0.10
0.11
d$r
sqrt(d$var.r)
Some additional information not expected in your answer:
However, this turns out to be a more a diﬃcult problem. If you use the percentilet bootstrap,
you are assuming that the t statistic is a pivotal quantit. That is, the tstatistic has the same
1
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Homework 7 Solution
Spring 2011
distribution no matter what the true value of the unknown parameters. That often happens
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