Exercise 6.9.2 Maximizing a Quadratic Response. Consider themodel, yi = 0+1xi+2x2 i +ei , ei s i.i.d.

Exercise 6.9.2 Maximizing a Quadratic Response.

Consider themodel, yi = β0+β1xi+β2x2 i

+ei , ei s i.i.d. N(0, σ2), i = 1, 2, 3, . . . , n.

Let x0 be the value at which the function E(y) = β0 +β1x +β2x2 is maximized (or minimized).

(a) Find the maximum likelihood estimate of x0.

(b) Find a (1−α)100% confidence interval for x0. Does such an interval always exist?

Hint: Use an F(1, n − 3) distribution based on ( ˆ β1 + 2 ˆ β2x0)2.

Comment: The problem of finding values of the independent variables that maximize

(or minimize) the expected y value is a basic problem in the field of response surface methods. See Box et al. (1978) or Christensen (2001, Chapter 8) or http://

www.stat.unm.edu/~fletcher/TopicsInDesign) for an introduction to the subject. Box and Draper (1987) give a detailed treatment.