384 Chapter 9: Regression
TABLE 9.3
xPPPˆ −Pˆ
5 .061 .063 −.002
10 .113 .109 .040
20 .192 .193 −.001
30 .259 .269 −.010
40 .339 .339 .000
50 .401 .401 .000
60 .461 .458 .003
80 .551 .556 −.005
Transforming this back into the original variable gives that the estimates ofcanddare
ˆc=e−A=.9847
1 −dˆ=e−B=.9901
and so the estimated functional relationship is
Pˆ= 1 −.9847(.9901)x
The residualsP−Pˆare presented in Table 9.3. ■
9.8Weighted Least Squares
In the regression model
Y=α+βx+e
it often turns out that the variance of a response is not constant but rather depends on its
input level. If these variances are known — at least up to a proportionality constant —
then the regression parametersαandβshould be estimated by minimizing a weighted
sum of squares. Specifically, if
Var(Yi)=
σ^2
wi
then the estimatorsAandBshould be chosen to minimize
∑
i
[Yi−(A+Bxi)]^2
Var(Yi)
=
1
σ^2
∑
i
wi(Yi−A−Bxi)^2