1
Sets, a reminder
It is beyond this course to dene a set in a precise way.
Cantor (1895) explains a set the following way: Unter einer Menge verstehen wir jede
Zusammenfssung M von bestimmten wohlunterschiedenen Objekten m unserer Anschauung oder unseres
Summary of Univariate normal statististical models. Denition
1
Let N be a nite set. The cardinality of N is also denoted N . Let L RN be a (linear)
subspace of the vector space RN . The statistical model
(N,2InN P(RN )|(, 2) L R+).
(1)
is called a (univar
The multivariate linear normal model (Testing)
1
Let L0 and L with L0 L RN be subspacces of RN and consider the two multivariate
linear normal models given by L0 and the nite set I and by L and the nite set I, i.e.,
the two statistical models:
H0 : (N0,Di
1
Univariate normal regression models.
It is seen that a univariate linear normal model
(N,2InN P(RN )|(, 2) L R+)
(1)
(and its statistical analysis) is determined solely by the subspace L RN . In many
univariate linear normal model the subspace is given
1
ML estimation of the multivariate variance.
Lemma: Let S P0(I). The maximum of the function
P(I) R+
1
|N/2
1
expcfw_ 2 Tr(1S)
exists if and only if S P(I) and in that case the maximizer is uniquely given by
1
S.
N
Proof: Suppose S P(I), i.e., is not s
1
Multivariate linear normal statistical models. The i.i.d. example.
We willl begin with a simple example of a multivariate linear normal statistical model.
Let I and N be nite sets. The i.i.d. Multivariate linear normal statistical model is
the statistic
1
Testing block independence, two blocks.
Consider the MANOVA model given by the subspace L RN and the nite set I (the
index set for the variates), i.e., the model
(N,Diag(|N ) P(RN I )|(, ) LI P(I).
We already know the complete solution to this model: If
Steen Andersson
16JAN07
M468
SOLUTIONS TO HOMEWORK 1
DUE THURSDAY 25JAN07
Problem 1. Let (Ni |i I) be a family of nite sets indexed by i I. Consider
the disjoint union (Ni |i I). An element of this set should naturally be denoted
i , indicating that the e
1
Matrices.
Let I, J and K be non-empty nite sets. A vector X in RIJ is also called an I J
matrix (with real valued entries). Thus X (X(i,j)|(i, j) I J ). Usually we
abbreviate with the notation Xij := X(i,j).
The transposed X t of a an I J matrix X RIJ i
Classical continuous distributions on R.
1
Gamma distributions: For any > 0 and > 0 the function f given by
x
1
1
x
expcfw_ , x > 0
f (x) :=
()
where x > 0 means that the function is ZERO for other x R, i.e., x 0, and ()
is the Gamma function: () :=
0
x1
Steen Andersson
21JAN07
M468
SOLUTION TO HOMEWORK 2
DUE THURSDAY 01FEB07
Problem 1. Using R to investigate the Beta distributions. For , R+ the Beta
distribution with parameters and is a probability distribution on the open unit
interval ]0, 1[ that we de
Steen Andersson
29JAN07
M468
SOLUTIONS TO HOMEWORK 3
DUE THURSDAY 08FEB07
Problem 1. Let (Nf |f F ) be a family of nite sets indexed by f F . Consider
the disjoint union N := (Nf |f F ). Thus N is a disjoint union of the subsets
Nf N , f F ) as mentioned
Steen Andersson
21JAN07
M468
HOMEWORK 2
DUE THURSDAY 01FEB07
Problem 1. Using R to investigate the Beta distributions. For , R+ the Beta
distribution with parameters and is a probability distribution on the open unit
interval ]0, 1[ that we denote by B( ,
Steen Andersson
16JAN07
M468
HOMEWORK 1
DUE THURSDAY 25JAN07
Problem 1. Let (Ni |i I) be a family of nite sets indexed by i I. Consider
the disjoint union (Ni |i I). An element of this set should naturally be denoted
i , indicating that the element i Ni ,