4.11 Consider a hot deck like that of Example 4.8, except that imputations are by random sampling of donors without replacement. To define the procedure when there are fewer donors than recipients, write n−r = kr + t, where k is a nonnegative integer and 0

NR = (kryR + tyt)∕(n − r), where yt is the mean of the t supplementary values of Y. If yHD2 denotes the estimate of Y from this procedure, then show that E( yHD2 ∣ Y(0)) = yR and Var( yHD2 ∣ Y(0)) = (t∕n)(1 − t∕r)s2 R∕n.

Show that the proportionate variance increase of yHD2 over yR is atmost 0.125, and this maximum is attained when k = 0, t = n/4, and r = 3n/4.