Name: _Solutions ________________________________
22S:166 Computing in Statistics
Instructor: Cowles
Midterm 2, 2005
Instructions: You may use Windows R, or if you prefer, you may log into a Linux
computer and run Linux R. Copy and paste your solutions into this Word document.
Email the document as an attachment in a message to me at
kate-cowles@stat.uiowa.edu.
1. The .75 quantile of a vector of sample values may be used to estimate the
population .75 quantile of the variable of interest.
a. Write an R function to get jackknife estimates of bias and standard error of this
estimator.
Copy your R code here.
> My.Jack
function(x)
{
n <- length(x)
theta.hatn <- fivenum(x)[4]
theta.hat <- rep(0,n)
theta.wig <- rep(0,n)
for(ii in 1:n)
{
theta.hat[ii] <- fivenum(x[-ii])[4]
theta.wig[ii] <- n * theta.hatn - (n-1) * theta.hat[ii]
}
thetahat.dot <- mean(theta.hat)
thetawig.dot <- mean(theta.wig)
se.thetan <- sqrt(sum((theta.wig - thetawig.dot)^2) / (n * (n-1)))
bias = (n-1) * (thetahat.dot - theta.hatn)
unbiased <- n * theta.hatn - (n-1) * thetahat.dot
list(Theta.hatn = theta.hatn, Theta.hat = theta.hat, Theta.wig = theta.wig,
Std.Error = se.thetan, Bias = bias, Theta.Jack = thetawig.dot)
}
b. Generate a vector of 20 random values from a Gamma distribution with
parameters 15 and 0.5. Use this vector as input to your function in part a.
Copy the R output here.
> My.Jack(rgamma(20,15,0.5))
$Theta.hatn
[1] 37.08957
$Theta.hat
