This problem introduces a technique called the “jackknife,” originally proposed by (1956) for reducing bias. Many nonlinear estimates, including the ratio estimator, have the property that
where θˆ is an estimate of θ. The jackknife forms an estimate , which has a leading bias term of the order n−2 rather than n−1. Thus, for sufficiently large n, the bias of is substantially smaller than that of θˆ. The technique involves splitting the sample into several subsamples, computing the estimate for each subsample, and then combining the several estimates. The sample is split into p groups of size m, where n = . For j = 1,…, p, the estimate is calculated from the m(p − 1) observations left after the group has been deleted. From the preceding expression,
Now, p “” are defined:
The jackknife estimate, , is defined as the average of the pseudovalues:
Show that the bias of is of the order n−2.