Page 116 - Applied Statistics with R

116 CHAPTER 7. SIMPLE LINEAR REGRESSION

2
where ∼ (0, ).

Then we can find the mean and variance of each .

E[ ∣ = ] = +

0
1
and
2
Var[ ∣ = ] = .

Additionally, the follow a normal distribution conditioned on the .

2
∣ ∼ ( + , )

1
0
2
Recall that the pdf of a random variable ∼ ( , ) is given by
1 1 − 2
2
( ; , ) = √ exp [− ( ) ].

2 2 2
Then we can write the pdf of each of the as

1 1 − ( + ) 2
2
( ; , , , ) = √ exp [− ( 0 1 ) ].
0 1 2 2
2
Given data points ( , ) we can write the likelihood, which is a function of

2
the three parameters , , and . Since the data have been observed, we use
0
1
lower case to denote that these values are no longer random.

1 1 − − 2
2
( , , ) = ∏ √ exp [− ( 0 1 ) ]
1
0
=1 2 2 2
2
Our goal is to find values of , , and which maximize this function, which
1
0
is a straightforward multivariate calculus problem.
We’ll start by doing a bit of rearranging to make our task easier.

1 1
2
2
( , , ) = (√ ) exp [− ∑( − − ) ]
1

0
1
0
2 2 2 2 =1
Then, as is often the case when finding MLEs, for mathematical convenience we
will take the natural logarithm of the likelihood function since log is a monoton-
ically increasing function. Then we will proceed to maximize the log-likelihood,
and the resulting estimates will be the same as if we had not taken the log.