Model Selection. Both selection criteria (2.19) and (2.20) are derived from information theoretic arguments, based on the well-known Kullback-Leibler discrimination information numbers (see Kullback and Leibler, 1951, Kullback, 1958). We give an argument due to Hurvich and Tsai (1989). We think of the measure (2.58) as measuring the discrepancy between the two densities, characterized by the parameter values θ0 1 = (β0 1, σ2 1)0 and θ0 2 = (β0 2, σ2 2)0

. Now, if the true value of the parameter vector is θ1, we argue that the best model would be one that minimizes the discrepancy between the theoretical value and the sample, say I(θ1;bθ). Because θ1 will not be known, Hurvich and Tsai

(1989) considered finding an unbiased estimator for E1[I(β1, σ2 1; βˆ,σˆ2)], where I(β1, σ2 1; βˆ,σˆ2) = 1 2

σ2 1

σˆ2 − log σ2 1

σˆ2 − 1



+

1 2

(β1 − βˆ)0 Z0 Z(β1 − βˆ)

nσˆ2 and β is a k × 1 regression vector. Show that E1[I(β1, σ2 1; βˆ,σˆ2)] = 1 2



− log σ2 1 + E1 log σb2 +

n + k n − k − 2 − 1



, (2.59)

using the distributional properties of the regression coefficients and error variance. An unbiased estimator for E1 log σb2 is log σb2. Hence, we have shown that the expectation of the above discrimination information is as claimed.

As models with differing dimensions k are considered, only the second and third terms in (2.59) will vary and we only need unbiased estimators for those two terms. This gives the form of AICc quoted in (2.20) in the chapter. You will need the two distributional results nσb2

σ2 1

∼ χ2 n−k and (βb − β1)0 Z0 Z(βb − β1)

σ2 1

∼ χ2 k

The two quantities are distributed independently as chi-squared distributions with the indicated degrees of freedom. If x ∼ χ2 n, E(1/x) = 1/(n − 2).