DUE FRIDAY 19JAN06
Problem 1. Consider the following three densities f1 , f2 , and f3 on R2 (wrt the
standard Lebesgue measure on R2 ):
f1 (x, y) = c1 expcfw_ [(x 1)2 + (y 2)2 ]
f2 (x, y) = c1 expcfw_ [x2 +
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.
UNIVARIATE LINEAR STATISTICAL MODELS, A SUMMARY
1. The canonical linear model, estimation
We refer to the lecture note on a summary of the multivariate normal distributions
for notation and fundament
THE MULTIVARIATE NORMAL DISTRIBUTION ON RI ,
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
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,
DUE THURSDAY 11APR06
Problem 1. Consider the statistical model (P P(NN )| R+ ), where P
denotes the Poisson distribution on N0 with parameter . Let the Gamma distribution = G, with shape parameter > 0 and scale
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
DUE TUESDAY 28FEB06
Problem. Consider the classical normal model
(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
DUE THURSDAY 09FEB06
Problem 1. State examples of a vector space V and a nite measure on V
a) = ,
b) = V .
c) Aspan() = V and = Int( ) V ,
d) Aspan() V and Int( ) = .
Problem 2. Let the measure on R be gi
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
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.
0% 25% 50%
926.0263 427.2295 69.30578 486 374 656 845 1142 2433 38
t = 13.361, df = 37, p-value = 9.895e-16
Alternative hypothesis: true mean is not equal to 0
95 percent confi