Leave-one-out jackknife

Quenouille (1949), Tukey (1958): the n leave-one-out values, bias (n-1)(mean(loo) - estimate) and variance (n-1)/n * sum((loo - mean(loo))^2). The (n-1) factors ARE the estimator – the leave-one-out values huddle n-1 times closer than independent replicates, and the inflation undoes exactly that; for the mean the jackknife variance equals s^2/n EXACTLY. Inconsistent for non-smooth statistics (the median is the canonical failure, Efron 1979 Sec. 3), and the output says so.

Usage

morie_bt_jackknife(x, stat)

Arguments

x

sample.

stat

the statistic; must be smooth for the variance to mean anything.

Value

list: leave_one_out, estimate, bias, corrected, variance, se, pseudovalues, n, smoothness_caveat, method.

References

Quenouille (1949), JRSS-B 11:68-84; Tukey (1958); Efron (1979), Sec. 3.

Examples

set.seed(1)
morie_bt_jackknife(stats::rnorm(30), mean)$variance
#> [1] 0.02846664