Homework 6 Solutions
(i) Note that l(x1 , ) ln p(x1 , ) = T (x1 ) A( ) + ln g (x1 ) l (x1 , ) = T (x1 ) A ( ) l (x1 , ) = A ( ). Since E [l (X1 , )] = 0, E T (X1 ) = A ( ). Also, I ( ) = E [l (X1 , )] = A ( ) > 0. Then, Var (T (X1 ) = E [(T (X1 ) A ( )2 ]
Statistics 610: Homework 5
Moulinath Banerjee University of Michigan Announcement: The Homework carries 120 points. Max possible score is 120. Due December 1. (1) FInd a test function that maximizes E (X ) subject to E (X 2 ) = E (1 X 2 ) = 1/2 invoking t
Statistics 610: Homework 1
September 16, 2008
Announcement: The homework carries 140 points but the maximum you can score is 110 points. (1) Recall the DemocratRepublican experiment we discussed in class. There are N individuals in a town marked by number
Statistics 610 Homework 1 Solutions
(1) Without loss of generality, let the Democrats be labeled 1 through m and the Republicans be labeled m + 1 through N . The number of permutations ( s) for which X1 = 1 is m (N 1)!, since the rst position can be lled
Statistics 610: Homework 3
(1) Problems from Keeners notes: Chapter 5. 14, 15, 18, 19, 20, 21, 25, 40. (2) (a) Independence of X and s. Let X1 , X2 , . . . , Xn be i.i.d. N (, 2 ). Using the method of orthogonal transformations discussed in class, show th
Parametric Inference
Moulinath Banerjee
University of Michigan April 14, 2004
1
General Discussion
The object of statistical inference is to glean information about an underlying population based on a sample collected from it. The actual population is ass
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STAT 610: STATISTICAL INFERENCE, FALL 2007
1
Instructor and Course Information
Instructor: Moulinath Banerjee Ofce: 451, West Hall Email: [email protected] Course Page: http:/www.stat.lsa.umich.edu/moulib/stat610.html Ofce Hours: Monday and Wednesday, 2:0
A Note on the Exponential Distribution
January 15, 2007
The exponential distribution is an example of a continuous distribution. A random variable X is said to follow the exponential distribution with parameter if its distribution function F is given by:
Exam 1 Solutions
(1) (a) The joint density of W1 , W2 , . . . , Wn is fa (w1 , w2 , . . . , wn ) = = = 1 2a 1 2a 1 2a
nn
1cfw_awi a
i=1 n
1cfw_amin wi max wi a
n
1cfw_maxcfw_|w(1) |,|w(n) |a ,
where w(1) = mini wi , and w(n) = maxi wi . Thus, by the facto
Homework 4 Solutions
(1) Recall that t-pivot is n(X ) tn1 , s
where s =
n 2 i=1 (Xi
X )2 /(n 1). Now, 2 = 1 , and let F be the CDF of
t-distribution with df = n 1. Then, n(X ) 1 P F (1 ) F 1 (1 + 1 ) , s from which we obtain the condence interval s s X F
1
Chapter 1 Special Distributions
1. Special Distributions Bernoulli, binomial, geometric, and negative binomial Sampling with and without replacement; Hypergeometric Finite sample variance correction Poisson and an informal Poisson process Stationary and
1
Chapter 2 Some Basic Large Sample Theory
1. Modes of Convergence Convergence in distribution, d Convergence in probability, p Convergence almost surely, a.s. Convergence in rth mean, r 2. Classical Limit Theorems Weak and strong laws of large numbers Cl
1
Chapter 4 Ecient Likelihood Estimation and Related Tests
1. Maximum likelihood and ecient likelihood estimation 2. Likelihood ratio, Wald, and Rao (or score) tests 3. Examples 4. Consistency of Maximum Likelihood Estimates 5. The EM algorithm and relate
Homework 6
Announcement: The Homework carries a total of 50 points. Let X1 , X2 , . . . , Xn , . be i.i.d. observations from a one parameter exponential family model p(x, ) = exp( T (x) A( ) h(x). Let 0 be the true underlying value of the parameter. (i) F
9. That Kn B(R) for all n implies that n Kn B(R). Note that cfw_Kn is decreasing and
2 (Kn ) = ( 3 )n . Thus, by the continuity of measure, (K ) = lim (Kn ) = 0. n
23. Note that (y 1) (1) for y < 0
P(Y y ) =
2(1) + cfw_(y 1) (1) otherwise. Thus, the den
Statistics 610: Homework 4
Moulinath Banerjee University of Michigan Announcement: The Homework carries 120 points. Max possible score is 120. Due Monday, Nov 17. (1) Let X1 , X2 , . . . , Xn be i.i.d. N (, 2 ). Recall that we can construct a level 1 cond