Exercise 6.9.1 Calibration. Consider the regression model Y = X + e, e N(0, 2 I

Exercise 6.9.1 Calibration.

Consider the regression model Y = Xβ +

e, e ∼ N(0, σ2 I ) and suppose that we are interested in a future observation, say y0, that will be independent of Y and have mean x

0β. In previous work with this situation, y0 was not yet observed but the corresponding vector x0 was known. The calibration problem reverses these assumptions.

Suppose that we have observed y0 and wish to infer what the corresponding vector x0 might be.

A typical calibration problem might involve two methods of measuring some quantity: y, a cheap and easy method, and x, an expensive but very accurate method.

Data are obtained to establish the relationship between y and x. Having done this, future measurements are made with y and the calibration relationship is used to identify what the value of y really means. For example, in sterilizing canned food, x would be a direct measure of the heat absorbed into the can, while y might be the number of bacterial spores of a certain strain that are killed by the heat treatment.

(Obviously, one needs to be able to measure the number of spores in the can both before and after heating.)

Consider now the simplest calibration model, yi = β0 + β1xi + ei , ei s i.i.d.

N(0, σ2), i = 1, 2, 3, . . . , n. Suppose that y0 is observed and that we wish to estimate the corresponding value x0 (x0 is viewed as a parameter here).

(a) Find the MLEs of β0, β1, x0, and σ2.

Hint: This is a matter of showing that the obvious estimates are MLEs.

(b) Suppose now that a series of observations y01, . . . , y0r were taken, all of which correspond to the same x0. Find the MLEs of β0, β1, x0, and σ2.
Hint: Only the estimate of σ2 changes form.

(c) Based on one observation y0, find a (1−α)100% confidence interval for x0.
When does such an interval exist?
Hint: Use an F(1, n − 2) distribution based on (y0 − ˆ β0 − ˆ β1x0)2.
Comment: Aitchison and Dunsmore (1975) discuss calibration in considerable detail, including a comparison of different methods.