Resampling, the jackknife, and pseudo-observations

Resampling methods approximate the sampling distribution of a statistic or estimator. In essence, a sample taken from the population is treated as a population itself. A large number of new samples, or resamples, are taken from this “new population”, commonly with replacement, and within each of these resamples, the estimate of interest is re-obtained. A large number of these estimate replicates can then be used to construct the empirical sampling distribution from which confidence intervals, bias, and variance may be estimated. These methods are particularly advantageous for statistics or estimators for which no standard methods apply or are difficult to derive.

The jackknife is a popular resampling method, first introduced by Quenouille in 1949 as a method of bias estimation. In 1958, jackknifing was both named by Tukey and expanded to include variance estimation. A jackknife is a multipurpose tool, similar to a swiss army knife, that can get its user out of tricky situations. Efron later developed the arguably most popular resampling method, the bootstrap, in 1979 after being inspired by the jackknife.

In Efron’s (1982) book The jackknife, the bootstrap, and other resampling plans, he states,

Good simple ideas, of which the jackknife is a prime example, are our most precious intellectual commodity, so there is no need to apologize for the easy mathematical level.

Despite existing since the 1940’s, resampling methods were infeasible due to the computational power required to perform resampling and recalculate estimates many times. With today’s computing power, the uncomplicated yet powerful jackknife, and resampling methods more generally, should be a tool in every analyst’s toolbox.

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Pride and Probability

To celebrate pride month, my husband Ethan’s workplace Desire2Learn organized virtual Drag Queen BINGO hosted by the fabulous Astala Vista. Even within the confines of a Zoom meeting, Astala Vista put on a great show!

Astala Vista, previously a self-proclaimed cat *lady* of drag, now a cat *cougar* at 30, demonstrating her “roar” on Zoom!

To keep things interesting (and I’m sure to reduce the odds of winning to keep the show going), different BINGO patterns besides the traditional “5 across” BINGO were used. This included a “4 corners” BINGO and a “cover-all” BINGO. To obtain a cover-all BINGO, all numbers on a traditional 5 by 5 BINGO card must be called (noting that of the 25 spaces on the card, 1 is a free space).

Up until this point, probability had not entered the discussion between my husband and I. However, with the cover-all BINGO, Ethan began wondering how many draws it would take to call a cover-all BINGO.

I became quiet, and my husband thought he had perhaps annoyed me with all of his probability questions. In fact, I was thinking about how I could easily simulate the answer to his question (and the corresponding combinatorics answer)!

First, we need to randomly generate a BINGO card. A BINGO card features five columns, with five numbers each. The exception is the N column which features a FREE space, given to all players. The B column features the numbers 1 through 15, the I column 16 through 30, etc. The numbers in each column are drawn without replacement for each card.

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Simplifying U-statistic variance estimation with Sen’s structural components

Sen (1960) proved that U-statistics could be decomposed into identically distributed and asymptotically uncorrelated “structural components.”

The mean of these structural components is equivalent to the U-statistic and the variance of the structural components can be used to estimate the variance of the U-statistic, bypassing the need for often challenging derivation of conditional variance components.

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One, Two, U: Examples of common one- and two-sample U-statistics

My previous two blog posts revolved around derivation of the limiting distribution of U-statistics for one sample and multiple independent samples.

For derivation of the limiting distribution of a U-statistic for a single sample, check out Getting to know U: the asymptotic distribution of a single U-statistic.

For derivation of the limiting distribution of a U-statistic for multiple independent samples, check out Much Two U About Nothing: Extension of U-statistics to multiple independent samples.

The notation within these derivations can get quite complicated and it may be a bit unclear as to how to actually derive components of the limiting distribution.

In this blog post, I provide two examples of both common one-sample U-statistics (Variance, Kendall’s Tau) and two-sample U-statistics (Difference of two means, Wilcoxon Mann-Whitney rank-sum statistic) and derive their limiting distribution using our previously developed theory.

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