Take Home final, Due before Dec. 14, 1PM



I. We all know that an estimator of the population sd is the sample
sd. When the population is normal, we can test or construct 
confidence interval
for the true population sd by using the F-distribution.

But when the population is clearly not normal and sample size is
not very large, the assumption of F-distribution for the
ratio of sample var to the population var is questionable.

Bootstrap can be used to find the distribution, replacing the 
F-distribution, when testing or construct confidence interval.

From the data set [http://www.ms.uky.edu/~mai/sta662/data2.txt]

use nonparametric bootstrap to find the approx. distribution of the
ratio of sample sd to population sd [using a table, and using some plots],
and thus construct a 
90% confidence interval for the population sd.

Compare this to the conficence interval using the F-distribution,
[normal assumption].



II.
Given a data set that is [konwn to be] a random sample from a
Gamma distribution with unknown shape and scale parameters.

http://www.ms.uky.edu/~mai/sta662/gamma.txt

We saw a possible (method of moment) estimator of the shape in the 
handout. And a possible estimator (again method of moment)
of the scale is

\hat  s = (sample variance) / (sample mean).

Use the parametric bootstrap [that is we assume the knowledge that
true data is from a gamma distrbution] with bootstrap iteration > 2000

(0) find the approximate distribution of  sqrt(n) [\hat \alpha - \alpha]

(1) find the approximate joint distribution of the 

sqrt(n)[ \hat s - s] ; sqrt(n)[ \hat \alpha - \alpha] 

by giving a scatter plot of these random vectors. 

(2) What is the likely marginal distributions? 
plot the normal QQ plots to see how good is a normal approx.


Next, construct 90% confidence
intervals of the following type
[We however are only interested in the shape estimator.]

(3) bootstrap-t, without using the sdfun() 
[note: it is OK to use the boott( ) function, although it is nonparametric
rather than parametric]

(4) plain percentile  [Note: you need to use parametric bootstrap here]

(5) BC  and BCa  percentile confidence interval [Note: you need to use parametric bootstrap here]