Exercise 6.9.3 Two-Phase Linear Regression.

Consider the problem of sterilizing canned pudding. As the pudding is sterilized by a heat treatment, it is simultaneously cooked. If you have ever cooked pudding, you know that it starts out soupy and eventually thickens. That, dear reader, is the point of this little tale. Sterilization depends on the transfer of heat to the pudding and the rate of transfer depends on whether the pudding is soupy or gelatinous. On an appropriate scale, the heating curve is linear in each phase. The question is, “Where does the line change phases?”

Suppose that we have collected data (yi , xi ), i = 1, . . . , n+m, and that we know that the line changes phases between xn and xn+1. The model yi = β10 +β11xi +ei , ei s i.i.d. N(0, σ2), i = 1, . . . , n, applies to the first phase and the model yi =

β20 + β21xi + ei , ei s i.i.d. N(0, σ2), i = n + 1, . . . , n + m, applies to the second phase. Let γ be the value of x at which the lines intersect.

(a) Find estimates of β10, β11, β20, β21, σ2, and γ .

Hint: γ is a function of the other parameters.

(b) Find a (1−α)100% confidence interval for γ . Does such an interval always exist?

Hint: Use an F(1, n + m − 4) distribution based on


( ˆ β10 + ˆ β11γ ) − ( ˆ β20 + ˆ β21γ )
2 .
Comment: Hinkley (1969) has treated the more realistic problem in which it is not known between which xi values the intersection occurs.