4.11. Consider a hot deck like that of Example 4.7, 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 ¼ kr þ t, where k is a non-negative integer and 0 < t < r. The hot deck without replacement selects all the recorded units k times, and then selects t additional units randomly without replacement to yield the n  r values required for the missing data. Thus y
* NR ¼ ðkry
R þ ty
tÞ=ðn  rÞ; where y
t is the mean of the t supplementary values of Y. If y
HD2 denotes the estimate of Y
from this procedure, then show that Eðy
HD2jYobsÞ ¼ y and Var(HD2 Yobs)=(t/n)(1 t/r)s/n. Show that the proportionate variance increase of HD2 Over VR is at most 0.125, and this maximum is attained when k = 0, t=n/4, and r=3n/4.