Exercise 2.11.1 Prediction. Consider a regression model Y=X + e, e N(0, 2 I ) and

Exercise 2.11.1 Prediction. Consider a regression model Y=Xβ +

e, e ∼ N(0,

σ2 I ) and suppose that we want to predict the value of a future observation, say y0, that will be independent of Y and be distributed N(x

0β, σ2).

(a) Find the distribution of y0 − x

0

ˆ  β

MSE



1 + x

0(XX)−1x0

 .

(b) Find a 95% prediction interval for y0.

Hint: A prediction interval is similar to a confidence interval except that, rather than finding parameter values that are consistent with the data and themodel, one finds new observations y0 that are consistent with the data and the model as determined by an α level test.

(c) Let η ∈ (0, 0.5]. The 100ηth percentile of the distribution of y0 is, say,

γ (η) = x

0β + z(η)σ . (Note that z(η) is a negative number.) Find a (1 − α)100%

lower confidence bound for γ (η). In reference to the distribution of y0, this lower confidence bound is referred to as a lower η tolerance point with confidence coefficient

(1 − α)100%. For example, if η = 0.1, α = 0.05, and y0 is the octane value of a batch of gasoline manufactured under conditions x0, then we are 95% confident that no more than 10% of all batches produced under x0 will have an octane value below the tolerance point.

Hint: Use a noncentral t distribution based on x

0

ˆ β − γ (η).

Comment: For more detailed discussions of prediction and tolerance (and we all know that tolerance is a great virtue), see Geisser (1993), Aitchison and Dunsmore

(1975), and Guttman (1970).