Relation of unbiasedness and invariance. (i) If 0 is the unique (up to sets of measure 0)

Relation of unbiasedness and invariance.

(i) If δ0 is the unique (up to sets of measure 0) unbiased procedure with uniformly minimum risk, it is almost invariant.

(ii) If G¯ is transitive and G∗ commutative, and if among all invariant (almost invariant) procedures there exists a procedure δ0 with uniformly minimum risk, then it is unbiased.

(iii) That conclusion (ii) need not hold without the assumptions concerning G∗ and G¯

is shown by the problem of estimating the mean ξ of a normal distribution N(ξ, σ2)

with loss function (ξ − d)2/σ2. This remains invariant under the groups G1 : gx =

x +

b, −∞ < b < ∞ and G2 : gx = ax +

b, 0 < a < ∞, −∞ < b < ∞. The best invariant estimate relative to both groups is X, but there does not exist an estimate which is unbiased with respect to the given loss function.

[(i): This follows from the preceding problem and the fact that when δ is unbiased so is g∗δg−1.

(ii): It is the defining property of transitivity that given θ, θ there exists g¯ such that

θ = ¯gθ. Hence for any θ, θ

EθL(θ

, δ0(X)) = EθL(g¯θ, δ0(X)) = EθL(θ, g∗−1

δ0(X)).

Since G∗ is commutative, g∗−1δ0 is invariant, so that R(θ, g∗−1

δ0) ≥ R(θ, δ0) = EθL(θ, δ0(X)).]