1.11 (a) If two estimators 1, 2 have continuous symmetric densities fi(x ), i = 1,...

1.11

(a) If two estimators δ1, δ2 have continuous symmetric densities fi(x − θ), i = 1, 2, and f1(0) > f2(0), then P[|δ1 − θ| < c] > P[|δ2 − θ| < c] for some c > 0 and hence δ1 will be closer to θ than δ2 with respect to the measure (1.5).

(b) Let X, Y be independently distributed with common continuous symmetric density f , and let δ1 = X, δ2 = (X + Y )/2. The inequality in part

(a) will hold provided 2

f 2(x) dx