Spring 2007
Statistics 703/J703 - Introduction to Statistical Theory II
Tuesday / Thursday 4:00-5:15
203 BA Building
Course Website: http://www.stat.sc.edu/~habing/courses/703S07.html

Assigned Thursday 1/25 Read for Tuesday 1/30
Should have read 8.1-8.4		 #1 - Due for Thursday 2/1
703h1S07.pdf
Assigned Thursday 2/1 Read for Tuesday 2/6
Should have read through 8.5.1		 #2 - Due for Thursday 2/8
1) Find the MLE for the geometric distribution parameter p.

2) This problem uses the logistic regression code on the R-templates page.
a) Briefly explain why the code needs to use the lines defining s and m. Be sure to include any math needed to show why that is correct.
b) Give a reason why using the value b=0 would be a bad choice for an initial value estimate.
c) Use the resulting estimated logistic regression equation to estimate the probability of O-ring failure at a temperature of 29 degrees (the temperature at launch time on January 27, 1986).
d) Imagine you were attempting to use logistic regression on something where an increase in X corresponded to a decrease in Y (say good cholesterol level and chance of a heart attack). What is one way you could modify the the code or the data to allow you to estimate this. (Note that you have to change something because a CDF is increasing!)

Assigned Thursday 2/8 Read for Tuesday 2/13
Should have read all of 8.5		 #3- Due for Thursday 2/15
This assignment deals with using the geometric distribution to examine the data in problem 8 on page 315.
a) Find the formula for the asymptotic variance of the mle for p.
b) Estimate p-hatmle for this data.
c) Use the asymptotic distribution of the mle to construct an approximate 95-percent confidence interval for p from this data.
d) Use the parametric bootstrap with the mle to construct an approximate 95-percent confidence interval for p from this data. (Hint: sort(lhat.dist)[50] andsort(lhat.dist)[950])
Assigned Tuesday 2/27 Read for Thursday 3/1
Should have read 9.1-9.2		 #4- Due for Tuesday 3/6
Chapter 9: #7 and 8. Also, consider a sample of size 20 and the null hypothesis lambda=1. Find the possible alpha level closest to 0.05 without going exceeding it, and the corresponding rejection region.

Note: You can use R to make a table of Poisson values for a given parameter with code similar to:
lambda<-7
range<-0:30
poistable<- cbind(range,round(ppois(range,lambda),3),1-round(ppois(range-1,lambda),3))
colnames(poistable)<-c("x","P(X<=x)","P(X>=x)")
poistable

Assigned Thursday 3/9 Read for Tuesday 3/20
Should have read 9.3-9.4		 #5 - Due for Thursday 3/22
A new type of product is supposed to have a mean time until failure of at least 5 hours. A sample of size 8 produced first failure times of 1.6, 4.3, 4.7, 5.8, 6.3, 2.1, 8.5, and 3.2.
a) Assuming the failure times follow an exponential distribution, find the UMP test for H0:mean failure time=5 hours versus HA: mean failure time < 5 hours. (Remember that the expected value is 1/lambda).
b) Find the p-value for testing these hypotheses with the given data set. (Hint: pgamma)
c) Find the power for testing these hypotheses when the true mean failure time is 4 hours.
Assigned Thursday 4/19 Read for Tuesday 4/24
Should have read 9.5,
8.7-8.8, 10.1-10.2, 10.4.6		 #6 - Due for Thursday 4/26
Find the minimum variance that any unbiased estimator of lambda for an exponential distribution can have.

Chapter 8: #75 (Identify the natural parameters and sufficient statistic)

Chapter 9: #36