Math 439 - Linear Statistical Models
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Why study linear models?
Simplicity
Intepretability
Obviousness of use
By far the most commonly used models
To complete your statistics major
2 / 132
Modelling
In statistics, we typically have a response variab

Estimating 2
It can be shown that
E(D 2 ) = (n p) 2 ,
and therefore, our unbiased estimator for 2 is
S2 =
20 / 54
D2
.
np
We can also show that the residual vector and our estimators are
uncorrelated.
Recalling
B = (X T X )1 X T Y ,
and
E = Y X B,
it suce

Multiple comparisons
In the previous set up where we have categorised our data into k groups, it
is a very natural question to ask, which groups are dierent from each
other?
If there are just a few groups, then theres quite a few, but a manageable
number

Homework 3: Due Wednesday, September 23
1. Consider the pvc dataset in the faraway package.
(a) Using psize as a response variable, t a two way anova model including the interaction term. Give interpretations for your estimated model
parameters.
(b) What

Homework 5: Due Wednesday, October 7
1. Derive the weighted least squares normal equations for tting a simple linear
2
regression when i = kXi , where k is a constant.
2. Show that M-estimation is a special case of weighted least squares.
3. Tukeys biweig

Homework 1: Due Wednesday, September 2
1. Show that for the general linear model that includes a constant intercept term,
(that is a column of 1s in the design matrix), a consequence of the method of
least squares is that the residuals sum to 0. It will h

Math 439 HW1
Yiyao Luo
443434
EX1 Solution:
n
2
i=1 :
Note that
,then we get 0 =
n
2
i=1 (yi Xi ) ,
:
=
n
i=1
and
:
n
i=1
Xi (yi Xi ) =
:
n
2
:
i=1 (yi Xi )
= 2
n
i=1
Xi :i .
Since for every Xi the rst element is 1, then the rst element of
thus
n
i=1
Xi (

Homework 4: Due Wednesday, September 30
1. Download the the helicopter.csv data from:
http:/math.wustl.edu/~hgan/helicopter.csv
Fit a polynomial regression to the data with time as the response and length
as the explanatory variable, taking care to look a

Homework 2: Due 3pm Friday, September 11
Questions marked with a * should be done completely by hand. Otherwise you may
use R in your solutions.
1. Using the sat data set from the faraway package, t the model
math = + 1 expend + 2 ratio + 3 salary.
Test t

Homework 1: Due Wednesday, September 2
1. Show that for the general linear model, a consequence of the method of least
squares is that the residuals sum to 0. It will help to write the sum of the
residuals as, n (yi Xi )2 .
i=1
2. Show that if we assume o

Math 439 - Linear Statistical Models
1 / 54
Why study linear models?
Simplicity
Intepretability
Obviousness of use
By far the most commonly used models
To complete your statistics major
2 / 54
Modelling
In statistics, we typically have a response variable