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 ...
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 ...