Page 249 - Probability and Statistical Inference

226 4. Functions of Random Variables and Sampling Distribution

Again, let us consider a partitioned matrix A from (4.8.7) where P u×u and
S w×w are square matrices where u + w = n. Then, one has

Let us reconsider the partitioned matrix A n×n which was used in (4.8.9). Then,
we can write

Next, we summarize an important tool which helps us to find the maxi-
mum or minimum of a real valued function of two variables. The result
follows:
Suppose that f(x) is a real valued function of a two-dimensional variable x
2
= (x , x ) ∈ (a , b ) × (a , b ) ⊆ ℜ , having continuous second-order partial
1 2 1 1 2 2
derivatives and everywhere
in an open rectangle (a , b ) × (a , b ). Suppose also that for some point ξ =
1 1 2 2
(ξ , ξ ) ∈ (a , b ) × (a , b ), one has
1 2 1 1 2 2

Now, let us denote the matrix of the second-order partial derivatives

Then,