Suppose that for the model in Example 6.4.6, the inference to be made is an estimate of the mean . Let T(x) be the estimate

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.