The Pitman Estimator of Location (see Lehmann and Casella 1998 Section 3.1, or the original paper by Pitman 1939) is given by

where we observe a random sample X1,..., Xn from f(x - θ). Pitman showed that this estimator is the location-equivariant estimator with smallest mean squared error (that is, it minimizes (7.3.3)). The goals of this exercise are more modest.
a. Show that dp(X) is invariant with respect to the location group of Example 7.3.6.
b. Show that if f(x - θ) is n(θ, 1), then dp(X) = .
c. Show that if f(x - θ) is uniform(θ - 1/2,θ + 1/2), then dp(X) = 1/2 (X(1) + X(n)).

Transcribed Image Text:

SII S( - t) dt dp(X) S II, (- t) dt