Reading: 6.5, Chapter 15, Appendix A.
An informal method of selecting p, the model order, is based on
Note: adding more terms always increases R2.
Formal methods can be based on hypothesis tests. We can test and then, if we accept this test and then, if we accept that test and so on stopping when we first reject a hypothesis. This is ``backwards elimination''.
Justification: Unless there is no good reason to suppose that and so on.
Apparent conclusion in our example: p=5 is best; look at the P values in the SAS outputs.
Problems arising with that conclusion:
Question: What is distribution theory?
Answer: How to compute the ``distribution'' of an estimator, test or other statistic, T:
In this course we
The standard normal density is
Reminder: if X has density f(x) then
So:
implies
Next we compute the variance of Z remembering that
:
where
and
dv = -ze-z2/2 dz. We do integration
by parts and see that
v=e-z2/2 and
.
This gives
because the integral of the normal density is 1. We have thus shown
that
Definition: If then .
Note:
Definition: If
are independent N(0,1)
then
We can define and for vectors like Z as follows:
If X is a random vector of length n, say
then
Definition:
Definition: If M is a matrix then is a matrix whose ijth entry is .
So has ijth entry and diagonal entries .
In class I started discussion of the Normal distribution. I computed the mean and variance of a standard normal and then of . Here I will just show you a few more integrals:
The kth moment of a standard normal is
We can also compute the moment generating function of Z, that is,
Now if then so that the central moment of X, namely is 0 for odd k and for k even.
Similarly