Theory Archives • Page 2 of 4 • Statisticelle

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.

Continue reading Resampling, the jackknife, and pseudo-observations

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.

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

Much Two U About Nothing: Extension of U-statistics to multiple independent samples

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Review

Statistical functionals are any real-valued function of a distribution function $F$ , $\theta = T(F)$ . When $F$ is unknown, nonparametric estimation only requires that $F$ belong to a broad class of distribution functions $\mathcal{F}$ , typically subject only to mild restrictions such as continuity or existence of specific moments.

For a single independent and identically distributed random sample of size $n$ , $X_1, …, X_n \stackrel{i.i.d}{\sim} F$ , a statistical functional $\theta = T(F)$ is said to belong to the family of expectation functionals if:

$T(F)$ takes the form of an expectation of a function $\phi$ with respect to $F$ ,
$T(F) = \mathbb{E}_F~ \phi(X_1, …, X_a)$
$\phi(X_1, …, X_a)$ is a symmetric kernel of degree $a \leq n$ .

A kernel is symmetric if its arguments can be permuted without changing its value. For example, if the degree $a = 2$ , $\phi$ is symmetric if $\phi(x_1, x_2) = \phi(x_2, x_1)$ .

If $\theta = T(F)$ is an expecation functional and the class of distribution functions $\mathcal{F}$ is broad enough, an unbiased estimator of $\theta = T(F)$ can always be constructed. This estimator is known as a U-statistic and takes the form,

$U_n = \frac{1}{{n \choose a}} \mathop{\sum … \sum} \limits_{1 \leq i_1 < ... < i_a \leq n} \phi(X_{i_1}, ..., X_{i_a})$

such that $U_n$ is the average of $\phi$ evaluated at all ${n \choose a}$ distinct combinations of size $a$ from $X_1, …, X_n$ .

For more detail on expectation functionals and their estimators, check out my blog post U-, V-, and Dupree statistics.

Since each $X_i$ appears in more than one summand of $U_n$ , the central limit theorem cannot be used to derive the limiting distribution of $U_n$ as it is the sum of dependent terms. However, clever conditioning arguments can be used to show that $U_n$ is in fact asymptotically normal with mean

$\mathbb{E}_F~ U_n = \theta = T(F)$

and variance

$\text{Var}_F~U_n = \frac{a^2}{n} \sigma_1^{2}$

where

$\sigma_1^{2} = \text{Var}_F \Big[ \mathbb{E}_F [\phi(X_1, …, X_a)|X_1] \Big].$

The sketch of the proof is as follows:

Express the variance of $U_n$ in terms of the covariance of its summands,

$\text{Var}_{F}~ U_n = \frac{1}{{n \choose a}^2} \mathop{\sum \sum} \limits_{\substack{1 \leq i_1 < ... < i_{a} \leq n \\ 1 \leq j_1 < ... < j_{a} \leq n}} \text{Cov}\left[\phi(X_{i_1}, ..., X_{i_a}),~ \phi(X_{j_1}, ..., X_{j_a})\right].$

Recognize that if two terms share $c$ common elements such that,
$\text{Cov} [\phi(X_1, …, X_c, X_{c+1}, …, X_a), \phi(X_1, …, X_c, X'_{c+1}, …, X'_a)]$

conditioning on their $c$ shared elements will make the two terms independent.
For $0 \leq c \leq n$ , define
$\phi_c(X_1, …, X_c) = \mathbb{E}_F \Big[\phi(X_1, …, X_a) | X_1, …, X_c \Big]$

such that

$\mathbb{E}_F~ \phi_c(X_1, …, X_c) = \theta = T(F)$

and

$\sigma_{c}^2 = \text{Var}_{F}~ \phi_c(X_1, …, X_c).$

Note that when $c = 0$ , $\phi_0 = \theta$ and $\sigma_0^2 = 0$ , and when $c=a$ , $\phi_a = \phi(X_1, …, X_a)$ and $\sigma_a^2 = \text{Var}_F~\phi(X_1, …, X_a)$ .
Use the law of iterated expecation to demonstrate that
$\sigma^{2}_c = \text{Cov} [\phi(X_1, …, X_c, X_{c+1}, …, X_a), \phi(X_1, …, X_c, X'_{c+1}, …, X'_a)]$

and re-express $\text{Var}_{F}~U_n$ as the sum of the $\sigma_{c}^2$ ,

$\text{Var}_F~U_n = \frac{1}{{n \choose a}} \sum_{c=1}^{a} {a \choose c}{n-a \choose a-c} \sigma^{2}_c.$

Recognizing that the first variance term dominates for large $n$ , approximate $\text{Var}_F~ U_n$ as

$\text{Var}_F~U_n \sim \frac{a^2}{n} \sigma^{2}_1.$
Identify a surrogate $U^{*}_n$ that has the same mean and variance as $U_n-\theta$ but is the sum of independent terms,
$U_n^{*} = \sum_{i=1}^{n} \mathbb{E}_F [U_n - \theta|X_i]$

so that the central limit may be used to show

$\sqrt{n} U_n^{*} \rightarrow N(0, a^2 \sigma_1^2).$
Demonstrate that $U_n - \theta$ and $U_n^{*}$ converge in probability,
$\sqrt{n} \Big((U_n - \theta) - U_n^{*}\Big) \stackrel{P}{\rightarrow} 0$

and thus have the same limiting distribution so that

$\sqrt{n} (U_n - \theta) \rightarrow N(0, a^2 \sigma_1^2).$

For a walkthrough derivation of the limiting distribution of $U_n$ for a single sample, check out my blog post Getting to know U: the asymptotic distribution of a single U-statistic.

This blog post aims to provide an overview of the extension of kernels, expectation functionals, and the definition and distribution of U-statistics to multiple independent samples, with particular focus on the common two-sample scenario.

Continue reading Much Two U About Nothing: Extension of U-statistics to multiple independent samples