(Jackknife) (pdf)

The Jackknife ApproachJackknife estimators allow to correct for a bias and its statistical error. Themethod was introduced in the 1950s in papers by Quenouille and Tukey. Thejackknife method is recommended as the standard for error bar calculations. Inunbiased situations the jackknife and the usual error bars agree. Otherwise thejackknife estimates are improvements, so that one cannot loose. In particular, thejackknife method solved the question of error propagation elegantly and with littleefforts involved.The unbiased estimator of the expectation value ̂x isN∑x = 1 Ni=1x iNormally bias problems occur when one estimates a non-linear function of ̂x:̂f = f(̂x) . (1)1

Typically, the bias is of order 1/N:bias (f) = ̂f − 〈f〉 = a 1N + a 2N 2 + O( 1N 3) . (2)Unfortunately, we lost the ability to estimate the variance σ 2 (f) = σ 2 (f)/N viathe standard equations 2 (f) = 1 N s2 (f) =1N (N − 1)N∑(f i − f) 2 , (3)i=1because f i = f(x i ) is not a valid estimator of ̂f. Also it is in non-trivial applicationsalmost always a bad idea to to use standard error propagation formulas with theaim to deduce △f from △x. Jackknife methods are not only easier to implement,but also more precise and far more robust.The error bar problem for the estimator f is conveniently overcome by using2

jackknife estimators f J , f J i, defined byf J = 1 NN∑fi J with fi J = f(x J i ) and x J i =i=11N − 1∑x k . (4)k≠iThe estimator for the variance σ 2 (f J ) iss 2 J(f J ) = N − 1NN∑(fi J − f J ) 2 . (5)i=1Straightforward algebra shows that in the unbiased case the estimator of thejackknife variance (5) reduces to the normal variance (3).Notably only of order N (not N 2 ) operations are needed to construct thejackknife averages x J i , i = 1, . . . , N from the original data. 3