Distribution Archives • Statisticelle

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.

Asymptotic distribution of U-statistics

One sample

For a single sample, $X_1, …, X_n \stackrel{i.i.d}{\sim} F$ , the U-statistic is given by

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

where $\phi$ is a symmetric kernel of degree $a$ .

For a review of what it means for $\phi$ to be symmetric, check out U-, V-, and Dupree Statistics.

In the examples covered by this blog post, $a = 2$ , so we can re-write $U_n$ as,

$U_n = \frac{1}{{n \choose 2}} \sum_{1 \leq i < j \leq n} \phi(X_{i}, X_{j}).$

Alternatively, this is equivalent to,

$U_n = \frac{1}{n(n-1)} \sum_{i \neq j} \phi(X_{i}, X_{j}).$

The limiting variance of $U_n$ is given by,

$Var ~U_n = \frac{1}{{n \choose a}} \sum_{c=1}^{a} {a \choose c}{n-a \choose a-c} \sigma_{c}^2$

where

$\sigma_{c}^2 = \text{Var}_F \Big[ \mathbb{E}_{F}~\Big( \phi(X_1, …, X_a) | X_1, …, X_c \Big)\Big] = \text{Var}_{F}~ \phi_c(X_1, …, X_c)$

or equivalently,

$\sigma_{c}^2 = \text{Cov} \Big[ \phi(X_1, …, X_c, X_{c+1}, …, X_a), \phi(X_1, …, X_c, X'_{c+1}, …, X'_a)\Big] .$

Note that when $a=c$ , $\phi_c(X_1, …, X_c) = \phi(x_1, …, x_a)$ .

For $a=2$ , these expressions reduce to

$\text{Var}_F~U_n = \frac{2}{n(n-1)} \left[2(n-2) \sigma_1^{2} + \sigma_2^{2} \right]$

where

$\sigma_{1}^{2} = \text{Var}_{F}~\phi_1(X_1) = \text{Var}_{F} \Big[ \mathbb{E}_{F} \Big( \phi(X_1, X_2) | X_1 \Big)\Big]$

and

$\sigma_{2}^{2} = \text{Var}_{F}~\phi(X_1, X_2).$

The limiting distribution of $U_n$ for $a=2$ is then,

$\sqrt{n}(U_n - \theta) \rightarrow N\left(0, \frac{2}{n-1} \left[2(n-2) \sigma_1^{2} + \sigma_2^{2} \right]\right).$

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.

Two independent samples

For two independent samples denoted $X_1, …, X_m \stackrel{i.i.d}{\sim} F$ and $Y_1, …, Y_n \stackrel{i.i.d}{\sim} G$ , the two-sample U-statistic is given by

$U = \frac{1}{{m \choose a}{n \choose b}} \mathop{\sum \sum} \limits_{\substack{1 \leq i_1 < ... < i_{a} \leq m \\ 1 \leq j_1 < ... < j_b \leq n}} \phi(X_{i_1}, ..., X_{i_a}; Y_{j_1}, ..., Y_{j_b}).$

where $\phi$ is a kernel that is independently symmetric within the two blocks $(X_{i_1}, ..., X_{i_a})$ and $(Y_{j_1}, ..., Y_{j_b})$ .

In the examples covered by this blog post, $a = b = 1$ , reducing the U-statistic to,

$U = \frac{1}{mn} \sum_{i=1}^{m} \sum_{j=1}^{n} \phi(X_i; Y_j) .$

The limiting variance of $U$ is given by,

$\text{Var}~U = \frac{a^2}{m} \sigma_{10}^2 + \frac{b^2}{n} \sigma_{01}^{2}$

where

$\sigma^{2}_{10} = \text{Cov} [\phi(X_1, X_2, …, X_a; Y_1, Y_2, …, Y_b), \phi(X_1, X'_2, …, X'_a; Y'_1, Y'_2…, Y'_b) ]$

and

$\sigma^{2}_{01} = \text{Cov} [\phi(X_1, X_2, …, X_a; Y_1, Y_2, …, Y_b),\phi(X'_1, X'_2, …, X'_a; Y_1, Y'_2…, Y'_b) ].$

Equivalently,

$\sigma^{2}_{10} = \text{Var} \Big[ \mathbb{E} \Big(\phi(X_1, …, X_a; Y_1, …., Y_b) | X_1 \Big) \Big] = \text{Var}~\phi_{10}(X_1)$

and

$\sigma^{2}_{01} = \text{Var} \Big[ \mathbb{E} \Big(\phi(X_1, …, X_a; Y_1, …, Y_b) | Y_1 \Big) \Big] = \text{Var}~\phi_{01}(Y_1).$

For $a=b=1$ , these expressions reduce to

$\text{Var}~U = \frac{\sigma_{10}^2}{m} + \frac{ \sigma_{01}^{2}}{n}$

where

$\sigma^{2}_{10} = \text{Cov} [\phi(X_1; Y_1), \phi(X_1; Y'_1) ] =\text{Var} \Big[ \mathbb{E} \Big(\phi(X_1; Y_1) | X_1 \Big) \Big]$

and

$\sigma^{2}_{01} = \text{Cov} [\phi(X_1; Y_1), \phi(X'_1; Y_1) ] =\text{Var} \Big[ \mathbb{E} \Big(\phi(X_1; Y_1) | Y_1 \Big) \Big] .$

The limiting distribution of $U_n$ for $a=b=1$ and $N=n+m$ is then,

$\sqrt{N}(U_n - \theta) \rightarrow N\left(0, \frac{N}{m} \sigma_{10}^2 + \frac{N}{n} \sigma_{01}^{2} \right).$

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.

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

Thank you very much to the lovely Feben Alemu for pointing me in the direction of https://pungenerator.org/ as a means of ensuring we never have to go without a brilliant title! With great power comes great responsibility.

Season 2 Crying GIF by Pose FX

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

Getting to know U: the asymptotic distribution of a single U-statistic

After my last grand slam title, U-, V-, and Dupree statistics I was really feeling the pressure to keep my title game strong. Thank you to my wonderful friend Steve Lee for suggesting this beautiful title.

Overview

A statistical functional is any real-valued function of a distribution function $F$ such that

$\theta = T(F)$

and represents characteristics of the distribution $F$ and include the mean, variance, and quantiles.

Often times $F$ is unknown but is assumed to belong to a broad class of distribution functions $\mathcal{F}$ subject only to mild restrictions such as continuity or existence of specific moments.

A random sample $X_1, …, X_n \stackrel{i.i.d}{\sim} F$ can be used to construct the empirical cumulative distribution function (ECDF) $\hat{F}_n$ ,

$\hat{F}_{n}(x) = \frac{1}{n} \sum_{i=1}^{n} \mathbb{I}(X_i \leq x)$

which assigns mass $\frac{1}{n}$ to each $X_i$ .

$\hat{F}_{n}$ is a valid, discrete CDF which can be substituted for $F$ to obtain $\hat{\theta} = T(\hat{F}_n)$ . These estimators are referred to as plug-in estimators for obvious reasons.

For more details on statistical functionals and plug-in estimators, you can check out my blog post Plug-in estimators of statistical functionals!

Many statistical functionals take the form of an expectation of a real-valued function $\phi$ with respect to $F$ such that for $a \leq n$ ,

$\theta = T(F) = \mathbb{E}_{F}~ \phi(X_1, …, X_a) .$

When $\phi(x_1, …, x_a)$ is a function symmetric in its arguments such that, for e.g. $\phi(x_1, x_2) = \phi(x_2, x_1)$ , it is referred to as a symmetric kernel of degree $a$ . If $\phi$ is not symmetric, a symmetric equivalent $\phi^{*}$ can always be found,

$\phi^{*}(x_1, …, x_a) = \frac{1}{a!} \sum_{\pi ~\in~ \Pi} \phi(x_{\pi(1)}, …, x_{\pi(a)})$

where $\Pi$ represents the set of all permutations of the indices $1, …, a$ .

A statistical functional $\theta = T(F)$ belongs to a special family of expectation functionals when:

$T(F) = \mathbb{E}_F ~\phi(X_1, …, X_a)$ , and
$\phi(X_1, …, X_a)$ is a symmetric kernel of degree $a$ .

Plug-in estimators of expectation functionals are referred to as V-statistics and can be expressed explicitly as,

$V_n = \frac{1}{n^a} \sum_{i_1 = 1}^{n} … \sum_{i_a = 1}^{n} \phi(X_{i_1}, …, X_{i_a})$

so that $V_n$ is the average of $\phi$ evaluated at all possible permutations of size $a$ from $X_1, …, X_n$ . Since the $X_i$ can appear more than once within each summand, $V_n$ is generally biased.

By restricting the summands to distinct indices only an unbiased estimator known as a U-statistic arises. In fact, when the family of distributions $\mathcal{F}$ is large enough, it can be shown that a U-statistic can always be constructed for expectation functionals.

Since $\phi$ is symmetric, we can require that $1 \leq i_1 < ... < i_a \leq n$ , resulting in ${n \choose a}$ combinations of the subscripts $1, ..., a$ . The U-statistic is then the average of $\phi$ evaluated at all ${n \choose a}$ distinct combinations of $X_1, ..., X_n$ ,

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

While $i_j \neq i_k$ within each summand now, each $X_i$ still appears in multiple summands, suggesting that $U_n$ is the sum of correlated terms. As a result, the central limit theorem cannot be relied upon to determine the limiting distribution of $U_n$ .

For more details on expectation functionals and their estimators, you can check out my blog post U-, V-, and Dupree statistics!

This blog post provides a walk-through derivation of the limiting, or asymptotic, distribution of a single U-statistic $U_n$ .

Continue reading Getting to know U: the asymptotic distribution of a single U-statistic

Parametric vs. Nonparametric Approach to Estimations

Parametric statistics assume that the unknown CDF $F$ belongs to a family of CDFs characterized by a parameter (vector) $\theta$ . As the form of $F$ is assumed, the target of estimation is its parameters $\theta$ . Thus, all uncertainty about $F$ is comprised of uncertainty about its parameters. Parameters are estimated by $\hat{\theta}$ , and estimates are be substituted into the assumed distribution to conduct inference for the quantities of interest. If the assumed distribution $F$ is incorrect, inference may also be inaccurate, or trends in the data may be missed.

To demonstrate the parametric approach, consider $n = 100$ independent and identically distributed random variables $X_1, …, X_n$ generated from an exponential distribution with rate $\lambda = 2$ . Investigators wish to estimate the 75 $^{th}$ percentile and erroneously assume that their data is normally distributed. Thus, $F$ is assumed to be the Normal CDF but $\mu$ and $\sigma^2$ are unknown. The parameters $\mu$ and $\sigma$ are estimated in their typical way by $\bar{x}$ and $\sigma^2$ , respectively. Since the normal distribution belongs to the location-scale family, an estimate of the $p^{th}$ percentile is provided by,

$x_p = \bar{x} + s\Phi^{-1}(p)$

where $\Phi^{-1}$ is the standard normal quantile function, also known as the probit.

set.seed(12345)
library(tidyverse, quietly = T)

# Generate data from Exp(2)
x <- rexp(n = 100, rate = 2)

# True value of 75th percentile with rate = 2
true <- qexp(p = 0.75, rate = 2) 
true

## [1] 0.6931472

# Estimate mu and sigma
xbar <- mean(x)
s    <- sd(x)

# Estimate 75th percentile assuming mu = xbar and sigma = s
param_est <- xbar + s * qnorm(p = 0.75)
param_est

## [1] 0.8792925

The true value of the 75 $^{th}$ percentile of $\text{Exp}(2)$ is 0.69 while the parametric estimate is 0.88.

Nonparametric statistics make fewer distributions about the unknown distribution $F$ , requiring only mild assumptions such as continuity or the existence of specific moments. Instead of estimating parameters of $F$ , $F$ itself is the target of estimation. $F$ is commonly estimated by the empirical cumulative distribution function (ECDF) $\hat{F}$ ,

$\hat{F}(x) = \frac{1}{n} \sum_{i=1}^{n} \mathbb{I}(X_i \leq x).$

Any statistic that can be expressed as a function of the CDF, known as a statistical functional and denoted $\theta = T(F)$ , can be estimated by substituting $\hat{F}$ for $F$ . That is, plug-in estimators can be obtained as $\hat{\theta} = T(\hat{F})$ .

Continue reading Parametric vs. Nonparametric Approach to Estimations