360 Chapter 9: Regression
Consequently, the maximum likelihood estimators ofαandβare precisely the values ofα
andβthat minimize
∑n
i= 1 (yi−α−βxi)^2. That is, they are the least squares estimators.
Notation
If we let
SxY=
∑n
i= 1
(xi−x)(Yi−Y)=
∑n
i= 1
xiYi−nxY
Sxx=
∑n
i= 1
(xi−x)^2 =
∑n
i= 1
xi^2 −nx^2
SYY=
∑n
i= 1
(Yi−Y)^2 =
∑n
i= 1
Yi^2 −nY^2
then the least squares estimators can be expressed as
B=
SxY
Sxx
A=Y−Bx
The following computational identity forSSR, the sum of squares of the residuals, can
be established.
Computational Identity forSSR
SSR=
SxxSYY−S^2 xY
Sxx
(9.3.4)
The following proposition sums up the results of this section.
PROPOSITION 9.3.1 Suppose that the responsesYi,i=1,...,nare independent normal
random variables with meansα+βxi and common varianceσ^2. The least squares
estimators ofβandα
B=
SxY
Sxx
, A=Y−Bx
are distributed as follows:
A∼N
α,
σ^2
∑
i
xi^2
nSxx
B∼N(β,σ^2 /Sxx)