Delta and jackknife estimators with low bias for functions of binomial and multinomial parameters

An estimator is said to be of order  s>0s>0 if its bias has magnitude n−sn−s, where nn is the sample size. We give delta estimators and jackknife estimators of order four for smooth functions of the parameters of a multinomial distribution. An unbiased estimator is given for its density function. We also give a jackknife estimator of any order for smooth functions of the binomial parameter.The jackknife estimator of order ss has a simpler form than the delta estimator of order ss. On the other hand, the jackknife estimator, like the bootstrap, requires ∼ns−1∼ns−1 calculations while the delta estimator of order ss requires only ∼n∼n calculations.Examples include the log odds ratio, the survival function and the Shannon information or entropy.