Suppose that for the model in Example 6.4.6, the inference to be made is an estimate of the mean μ. Let T(x) be the estimate used if X = x is observed. If ga(X) = Y = y is observed, then let T*(y) be the estimate of μ + a, the mean of each Yi. If μ + a is estimated by T*(y), then μ would be estimated by T*(y) - a.
a. Show that measurement equivariance requires that T(x) = T*(y) - a for all x = (xi,... ,xn) and all a.
b. Show that formal invariance requires that T(x) = T*(x) and hence the Equivariance Principle requires that T(x1,...,xn) + a = T(x1 + a,... ,xn + a) for all (x1,..., xn) and all a.
c. If X1,... ,Xn are iid f(x - θ), show that, as long as E0X1 = 0, the estimator W(X1,..., Xn) =  is equivariant for estimating θ and satisfies EθW - θ.