1. The Multivariate Normal Distribution, Part 2
BMTRY 726
5/22/2018

2. More Properties of MVN Last lecture we discussed:
The form of the MVN distribution
Contours of constant density obtained by taking a slice of the MVN distribution as some set height
Some of the properties of the MVN distribution
Impact of linear combinations of X
Partitions of X
Conditions for Independence of vectors in X
We will continue this discussion with some additional useful properties

3. Conditional Distributions
Result 4.6: Suppose
Then the conditional distribution of X1 given that X2 = x2 is a normal distribution
Note the covariance matrix does not depend on the value of x2

4. Proof of Result 4.6

5. Proof of Result 4.6

6. Proof of Result 4.6

7. Example
Consider
Find the conditional distribution of the 1st and 3rd components

8. Example

9. Example

10. Results 4.6 & Multiple Regression
Consider
The conditional distribution of Y|X=x is univariate normal with

11. Result 4.7: If and S is positive definite, then
Proof:

12. Result 4.7: If and S is positive definite, then
Proof cont’d:

13. Result 4.8: If are mutually independent with
And c1, c2, …,cn are n constants.
Then
Additionally if we have and which are r x p matrices of constants we can also say

14. Sample Data Let’s say that X1, X2, …, Xn are i.i.d. random vectors
If the data vectors are sampled from a MVN distribution then

15. Multivariate Normal Likelihood
We can also look at the joint likelihood of our random sample

16. Some needed Results (1) Given A > 0 and are eigenvalues of A (a)
(b)
(c)
(2) From (c) we can show that:

17. Some needed Results
(2) Proof that:

18. Some needed Results
(2) Proof that:

19. Some needed Results
(2) Proof that:

20. Some needed Results (1) Given A > 0 and are eigenvalues of A (a)
(b)
(c)
(2) From (c) we can show that:
(3) Given Spxp > 0, Bpxp > 0 and scalar b > 0

21. MLE’s for

22. MLE’s for

23. MLE’s for

24. MLE’s for

25. A Few Notes About The MLE’s for Variance
As in the univariate setting, the MLE for the variance matrix is biased
Thus we generally use an alternative to the MLE…

26. Sampling Distributions
So we’ve discussed that we can estimate the mean vector, m, and the covariance matrix, S, using and S
But we need to understand how these are distributed..

27. Sample Mean Vector
We can estimate a sample mean for X1, X2, …, Xn

28. Sample Mean Vector Now we can estimate the mean of our sample
But what about the properties of ?
It is an unbiased estimate of the mean
It is a sufficient statistic
Also, the sampling distribution is:

29. .

30. .

31. Sample Covariance And the sample covariance for X1, X2, …, Xn
Sample variance
Sample Covariance

32. Sample Mean Vector So we can also estimate the variance of our sample
And like , S also has some nice properties
It is an unbiased estimate of the variance
It is also a sufficient statistic
It is also independent of
But what about the sampling distribution of S?

33. Wishart Distribution
Given , the distribution of is called a Wishart distribution with n degrees of freedom.
has a Wishart distribution with n -1 degrees of freedom
The density function is
where A and S are positive definite

34. Wishart cont’d
The Wishart distribution is the multivariate analog of the central chi-squared distribution.
If are independent then
If then CAC’ is distributed
The distribution of the (i, i) element of A is

35. Large Sample Behavior
Let X1, X2, …, Xn be a random sample from a population with mean and variance (not necessarily normally distributed)
Then and S are consistent estimators for m and S. This means

36. Large Sample Behavior
If we have a random sample X1, X2, …, Xn a population with mean and variance, we can apply the multivariate central limit theorem as well
The multivariate CLT says

37. Next Time Checking Normality
How can we check MVN and what do we do if our data don’t appear MVN?
SAS and R
Begin our discussion of statistical inference for MV vectors