UPenn, STAT 512
Excerpt: ... Homework 8, Statistics 512, Spring 2005 This homework is due Thursday, March 24th at the beginning of class. 1. Let X 1 ,K , X n be iid Bernoulli ( p ) random variables. (a) Find the size 0.05 Wald test of H 0 : p = 0.5 versus H a : p X 0.5 . (b) Find the size 0.05 likelihood ratio test of H 0 : p = 0.5 versus H a : p X 0.5 . (c) Conduct a Monte Carlo simulation study in R to compare the actual size of the tests in (a) and (b) for n = 20 , p = 0.5 . (d) Conduct a Monte Carlo simulation study in R to compare the power of the tests in (a) and (b) for n = 20 , p = 0.4 . 2. Hogg, McKean and Craig, Exercise 6.3.1. Hint: Review Section 3.3, particularly Theorem 3.3.2. 3. Hogg, McKean and Craig, Exercise 6.4.7 4. (a) Hogg, McKean and Craig, Exercise 6.4.8. (b) Hogg, McKean and Craig, Exercise 6.4.9. Note: Location-scale families are commonly used in statistics. Some examples are the following: (i) normal distribution (ii) double exponential distribution with a scale 1 -| x -a|/ b e parameter f ( x; a, b) = ; (iii) e ...
N.C. State, ST 745
Excerpt: ... nformation matrix from this model and then evaluate them under the hypothesis that the data are from an exponential distribution. (c) Perform the score test to test whether or not the survival times are from an exponential distribution. 5. Consider the AZT cohort study data of the textbook in problem 4.7.7 (Page 122, the data is also available from the online link of the book). Do the following by using statistical software (such as SAS or R): (a) Fit a Weibull model to the censored survival data (b) Perform Wald test to test whether or not the survival times are from an exponential distribution. (c) Perform likelihood ratio test to test whether or not the survival times are from an exponential distribution. (d) Suggest ways to check the Weibull model assumption and conduct the diagnostics. 2 ...
Washington University in St. Louis, MATH 5062
Excerpt: ... Washington University Math5062 (Spring 2007) 1 Homework 5: Due Wednesday March 28, 2006 1. Problem 6.3.1 on page 428 in BD. 2. Problem 6.3.6 on page 430 in BD. 3. Let X1 , . . . , Xn be i.i.d. N (, 1) with 0. And we want to test H0 : = 0 versus H1 : > 0. Note that since the parameter space is not open, the regular asymptotics do not apply. (a) Show that the MLE of is (b) If > 0, show that ^ n = Xn 1{Xn >0} . d ^ n(n - ) N (0, 1) ^ d ^ (c) If = 0, the probability is 1/2 that n = 0 and 1/2 that n(n - ) N (0, 1). (d) Derive the maximum likelihood ratio test at asymptotic level . ...
UCSB, PSTAT 120c
Excerpt: ... PSTAT 120C: Assignment # 1 These problems are due at the start of lecture on Thursday. Due April 13, 2006 1. Show that if T is a sufficient statistic for estimating from the data X1 , . . . , Xn then the Generalized Likelihood Ratio Test (GLRT) statistic for testing H0 : 0 versus Ha : 0 is a function of T . (Hint: use the factorization theorem.) 2. Suppose that X is a binomial random variable from n trials with probability p. We want to 1 test H0 : p = 2 versus Ha : p = 1 . 2 (a) Find , the GLRT statistic. (b) Argue that the critical region from the GLRT is of the same form as X- 3. Exercise 10.93 on page 523 in the textbook. 4. Exercises 8.81 and 8.87 on pages 409410 in the textbook. 5. Exercise 10.67 on pages 505506 in the textbook. n k 2 1 ...