Treating jackknife pseudovalues as IID variables is ubiquitous among the sources I've come across. However, I never see an attempt to justify it beyond citing a "proposal" by Tukey's 1958 paper, which I can't seem to find online.

The pseudovalues don't seem independent because they are all computed from the same sample; change one value in the sample, and you change all but one pseudovalue. We're not even randomly sampling the original sample. The sources all seem to tacitly acknowledge this by saying it's "approximately" or "treated as" independent. What gives?


2 Answers 2


Daniel Thornburn has a couple of relatively readable papers on this topic in Biometrika, 1976, vol 62, issue 2, pages 305-13, and Scand J Statist, 1977, vol 4, pages 113-8. He shows that under certain conditions the covariance between pseudovalues falls like 1/n. Therefore for a respectable sample size n, pseudovalues can be considered "almost independent".

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    $\begingroup$ +1 Thank you for bringing your scholarship to this question. $\endgroup$
    – whuber
    Jan 16, 2023 at 14:14

Here's what I think. The jackknife pseudovalues are defined as $x_i^*=n\theta_n-(n-1)\theta_{n-1;-i}$, where $\theta_n$ is the estimator based on all n observations and $\theta_{n-1;-i}$ is based on all observations except observation i. Surely these are identically distributed as long as $\theta_n$ is symmetric in the observations (you don't question this anyway).

Now if $\theta_n$ is the arithmetic mean of $x_1,\ldots,x_n$, $x_i^*=x_i$, iid if the original observations are iid, no problem. Same if $\theta_n$ is the arithmetic mean of some function of $x_i$. When showing asymptotic normality of many estimators, the way to do this is to approximate them by the arithmetic mean of some function of $x_i$, say $f(x_i)$, plus some remainder that vanishes for $n\to\infty$ and/or divided by something that converges to a constant $c$, so that the Central Limit Theorem can be applied. In which case the jackknife pseudovalues are approximately $f(x_i)/c$ (iid) plus something that is a function of the vanishing remainders.

Probably one could generally "approximate" estimators in some more or less optimal way by means of observation-wise contributions $f(x_i)$ making the pseudovalues iid, however without further assumptions this approximation may not be any good and may not be valid even for $n\to\infty$.


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