Steen Andersson
12JAN06
M567
HOMEWORK 1
DUE FRIDAY 19JAN06
Problem 1. Consider the following three densities f1 , f2 , and f3 on R2 (wrt the
standard Lebesgue measure on R2 ):
1
f1 (x, y) = c1 expcfw_ [(x 1)2 + (y 2)2 ]
1
2
1
f2 (x, y) = c1 expcfw_ [x2 +
Spring 2005
EXPONENTIAL FAMILIES
A DRAFT
BY
STEEN ANDERSSON
INDIANA UNIVERSITY
1. The multivariate Laplace transform.
Let V = cfw_0 be a nite dimensional vector space and V the dual vector space,
i.e., the vector space of all linear mappings from V to R.
Spring 2005
M 567
UNIVARIATE LINEAR STATISTICAL MODELS, A SUMMARY
BY
STEEN ANDERSSON
INDIANA UNIVERSITY
1. The canonical linear model, estimation
We refer to the lecture note on a summary of the multivariate normal distributions
for notation and fundament
Spring 2005
THE MULTIVARIATE NORMAL DISTRIBUTION ON RI ,
A SUMMARY
BY
STEEN ANDERSSON
INDIANA UNIVERSITY
1. Vectors, linear mappings, and bi-linear forms
Let I and J be a nite sets. The vector space RI is thus all families x (xi |i I)
of real numbers inde
Steen Andersson
07FEB06
M567
SOLUTIONS TO HOMEWORK 7
DUE THURSDAY 07MAR
Although this Homework 7 seems very long it is in fact very short. Essential you
are just asked to write down, test statistics, asymptotic distributions for estimators,
and asymptotic
Steen Andersson
30MAR06
M567
HOMEWORK 8
DUE THURSDAY 11APR06
N
Problem 1. Consider the statistical model (P P(NN )| R+ ), where P
0
denotes the Poisson distribution on N0 with parameter . Let the Gamma distribution = G, with shape parameter > 0 and scale
Steen Andersson
14FEB06
M567
HOMEWORK 5
DUE TUESDAY 21FEB06
Problem 1: For each of the following statistical models (P P()| ) below:
(i): Establish that the model is an exponential family.
(ii): Find a minimal representation of the model as an exponential
Steen Andersson
16FEB06
M567
HOMEWORK 6
DUE TUESDAY 28FEB06
Problem. Consider the classical normal model
(1)
(N (, 2 ) P(R)|(, 2 ) R R+ ),
where N (, 2 ) denotes the normal distribution on R with expectation and variance 2 .
a) Establish that the classica
Steen Andersson
02FEB06
M567
HOMEWORK 4
DUE THURSDAY 09FEB06
Problem 1. State examples of a vector space V and a nite measure on V
such that:
a) = ,
b) = V .
c) Aspan() = V and = Int( ) V ,
d) Aspan() V and Int( ) = .
Problem 2. Let the measure on R be gi
Steen Andersson
19JAN06
M567
HOMEWORK 2
DUE THURSDAY 26JAN06
On the linear regression model on RN , partly recap: Let D be an N F
matrix of full rank F < N . Let 2 denote the standard inner product on RN ,
i.e., x 2 := (x2 | N ), x (x | N ) RN . Consider
Steen Andersson
26MAR06
M567
HOMEWORK 3
DUE THURSDAY 02FEB06
On the linear normal model on RN , partly recap: Let D be the N F matrix
of rank F < N . Consider the regression normal model
(N (D, 2 1N ) P(RN )|(, 2 ) RF R+ ),
the hypothesis denoted H.
Let F
Lab 6
Karlie Gray
math 220
Question 1
A)
B) mean
sd
se(mean) IQR
0% 25% 50%
75% 100%
n
926.0263 427.2295 69.30578 486 374 656 845 1142 2433 38
C)
t = 13.361, df = 37, p-value = 9.895e-16
Alternative hypothesis: true mean is not equal to 0
95 percent confi