STAT 351
Confidence intervals for
Refer to the polynomial regression example
(data on insurance costs). I fit polynomials of
degree 1 through 5. Each model gives a
vector of fitted parameters and to predict
the mean value of Y at time t we use
when the fi
STAT 351
The kth moment of a standard normal is
We can do this integral by parts with
and
and
parts
Since
. This makes
so that by integration by
and
we see that
if k is
odd and
if k is even.
We can also compute the moment generating
function of Z, that is
STAT 351
Joint Densities
Suppose and are independent standard
normals. In class I said that their joint
density was
Here I want to show you the meaning of joint
density by computing the density of a
random variable.
Let
. By definition U has a
distributio
STAT 351
Another Extra Sum of Squares Example: two
way layout
We have data
for i from 1 to I, j from 1
to J and k from 1 to K where ilabels the row
effect, j labels the column effect and k labels
the replicate. When K is more than 1 we
generally check for
STAT 351
Here is an example of some of the matrix
algebra I was doing in class. Consider the
weighing design where two objects of
weights and are weighed individually and
together. The resulting design matrix is
So
and
The hat matrix is
which is
Notice th
STAT 351
Multiple Regression
Here is a page of plots:
You will see that the plot of residual against fitted value suggests that we should
consider adding a quadratic term in Fibre:
The model
has ESS=87.083 on 14 degrees of freedom while the model
has ESS
STAT 351
I tried, as I was calculating things for the
vectors , and to emphasize which things
needed which assumptions.
So for instance we have following matrix
identities which depend only on the model
equation
where is the `hat' matrix
,
If we add the a
STAT 351
Standardized Residuals
For the insurance data we can look at the residuals after various model fits.
data insure;
infile 'insure.dat' firstobs=2;
input year cost;
code = year - 1975.5 ;
proc glm data=insure;
model cost = code ;
output out=insfit
STAT 351
Reparametrization
Suppose X is the design matrix of a linear
model and that is the design matrix of the
linear model we get by imposing some linear
restrictions on the model using X. A good
example is on assignment 3 but here is
another. Consider
STAT 351
Multiple Regression
In the data set below the hardness of plaster
is measured for each of 9 combinations of
sand content and fibre content. Sand content
was set at one of 3 levels as was fibre content
and all possible combinations tried on two
ba
STAT 351
Polynomial Regression
We consider data on average claims paid per
policy for automobile insurance in New
Brunswick in the years 1971-1980:
Year 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980
Cost 45.13 51.71 60.17 64.83 65.24 65.17 67.65 79.80