The jackknife is a general technique for reducing bias in an estimator (Quenouille, 1956). A one-step jackknife estimator is defined as follows. Let X1,..., Xn be a random sample, and let Tn = Tn(X1,..., Xn) be some estimator of a parameter 6. In order to "jackknife" Tn we calculate the n statistics Tn i = 1,..., n, where Tn(i) is calculated just as Tn but using the n - 1 observations with Xi removed from the sample. The jackknife estimator of 8, denoted by JK(Tn), is given by

(In general, JK(Tn) will have a smaller bias than Tn. See Miller 1974 for a good review of the properties of the jackknife.)
Now, to be specific, let X1,..., Xn be iid Bernoulli(0). The object is to estimate 82.
(a) Show that the MLE of θ2, (ˆ‘ni=1Xi/n)2, is a biased estimator of θ2.
(b) Derive the one-step jackknife estimator based on the MLE.
(c) Show that the one-step jackknife estimator is an unbiased estimator of θ2. (In general, jackknifing only reduces bias. In this special case, however, it removes it entirely.)
(d) Is this jackknife estimator the best unbiased estimator of θ2? If so, prove it. If not, find the best unbiased estimator.

n 1