This exercise involves some details in Example 14.5. (i) Show that MSPE( ) = 1 B. (ii) Show that MSPE( ) = 1

This exercise involves some details in Example 14.5.

(i) Show that MSPE( ˜ζ) = 1 − B.

(ii) Show that MSPE( ˆζ) = 1 − B + 2B/m + o(m

−1).

(iii) Show that E( ˆ ζ

∗−

1

− ˆζ)2 = A(1 − B) + o(1).

(iv) Show that E( ˆ ζ

∗−

i

− ˆζ)2 = 2B/m2 + o(m

−2), i ≥ 2.

(v) Show that, with ˆ ζ−i replaced by ˆ ζ

∗−

i, 1 ≤ i ≤ m, the expectation of the right side of (14.27) is equal to MSPE( ˆζ) + A(1 − B) + o(1).