5.15 There are various ways to seemingly generalize Theorems 5.5 and 5.9. However, if both the estimator and loss function are allowed to depend on the covariance and loss matrix, then linear transformations can usually reduce the problem.

Let X ∼ Nr(θ , ), and let the loss function be L(θ , δ)=(θ −δ)



Q(θ −δ), and consider the following “generalizations” of Theorems 5.5 and 5.9.

(a) δ(x) = 

1 − c(x

−1x)

x

−1x



x, Q = −1

,

(b) δ(x) = 

1 − c(x

Qx)

x

Qx



x,  = I or  = Q,

(c) δ(x) = 

1 − c(x

−1/2Q−1/2x)

x

−1/2Q−1/2x



x.

In each case, use transformations to reduce the problem to that of Theorem 5.5 or 5.9, and deduce the condition for minimaxity of δ.

[Hint: For example, in

(a) the transformation Y = −1/2X will show that δ is minimax if 0 < c(·) < 2(r − 2).]