Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

9.6. A Regression Problem 549

9.6.6.Consider Model (9.6.1). Letη 0 =E(Y|x=x 0 −x). The least squares
estimator ofη 0 is ˆη 0 =ˆα+βˆ(x 0 −x).

(a)Using (9.6.9), show that ˆη 0 is an unbiased estimator and show that its variance

is given by

V(ˆη 0 )=σ^2

[

1

n

+

(x 0 −x)^2

∑n

i=1(x^1 −x)

2

]

(b)Obtain the distribution of ˆη 0 and use it to determine a (1−α)100% confidence

interval forη 0.

9.6.7.Assume that the sample (x 1 ,Y 1 ),...,(xn,Yn) follows the linear model (9.6.1).
SupposeY 0 is a future observation atx=x 0 −xand we want to determine a pre-
dictive interval for it. Assume that the model (9.6.1) holds forY 0 ; i.e.,Y 0 has a
N(α+β(x 0 −x),σ^2 ) distribution. We use ˆη 0 of Exercise 9.6.6 as our prediction of
Y 0.

(a)Obtain the distribution ofY 0 −ηˆ 0 , showing that its variance is:

V(Y 0 −ηˆ 0 )=σ^2

[

1+

1

n

+

(x 0 −x)^2

∑n

i=1(x^1 −x)

2

]

Use the fact that the future observationY 0 is independent of the sample

(x 1 ,Y 1 ),...,(xn,Yn).

(b)Determine at-statistic with numeratorY 0 −ηˆ 0.

(c)Now beginning with 1−α=P[−tα/ 2 ,n− 2 <T <tα/ 2 ,n− 2 ], where 0<α<1,

determine a (1−α)100% predictive interval forY 0.

(d)Compare this predictive interval with the confidence interval obtained in Ex-

ercise 9.6.6. Intuitively, why is the predictive interval larger?

9.6.8.In Example 9.6.1, we obtain the predicted winning time for the men’s 1500

meters in the 2020 Olympics. Compute the 95% predictive interval for this predic-

tion that is given in the last exercise. These computations are performed by the R

functioncipi.R. The call iscipi(lm(time~year),matrix(c(1,2020),ncol=2)).

In terms of the problem, what does this predictive interval mean? Next compute

the prediction for the 2024 and 2028 Olympics. Why are the intervals increasing in

length?

9.6.9.Show that

∑n

i=1

[Yi−α−β(xi−x)]^2 =n(ˆα−α)^2 +(βˆ−β)^2

∑n

i=1

(xi−x)^2 +

∑n

i=1

[Yi−αˆ−βˆ(xi−x)]^2.

9.6.10.Let the independent random variablesY 1 ,Y 2 ,...,Ynhave, respectively, the
probability density functionsN(βxi,γ^2 x^2 i),i=1, 2 ,...,n, where the given numbers
x 1 ,x 2 ,...,xnare not all equal and no one is zero. Find the maximum likelihood
estimators ofβandγ^2.