statistical functionals Archives • Statisticelle

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

U-, V-, and Dupree statistics

To start, I apologize for this blog’s title but I couldn’t resist referencing to the Owen Wilson classic You, Me, and Dupree – wow! The other gold-plated candidate was U-statistics and You. Please, please, hold your applause.

My previous blog post defined statistical functionals as any real-valued function of an unknown CDF, $T(F)$ , and explained how plug-in estimators could be constructed by substituting the empirical cumulative distribution function (ECDF) $\hat{F}_{n}$ for the unknown CDF $F$ . Plug-in estimators of the mean and variance were provided and used to demonstrate plug-in estimators’ potential to be biased.

$\hat{\mu} = \mathbb{E}_{\hat{F}_n}[X] = \sum_{i=1}^{n} X_i P(X = X_i) = \frac{1}{n} \sum_{i=1}^{n} X_i = \bar{X}_{n}$

$\hat{\sigma}^{2} = \mathbb{E}_{\hat{F}_{n}}[(X- \mathbb{E}_{\hat{F}_n}[X])^2] = \mathbb{E}_{\hat{F}_n}[(X - \bar{X}_{n})^2] = \frac{1}{n} \sum_{i=1}^{n} (X_i - \bar{X}_{n})^2.$

Statistical functionals that meet the following two criteria represent a special family of functionals known as expectation functionals:

1) $T(F)$ is the expectation of a function $g$ with respect to the distribution function $F$ ; and

$T(F) = \mathbb{E}_{F} ~g(X)$

2) the function $g(\cdot)$ takes the form of a symmetric kernel.

Expectation functionals encompass many common parameters and are well-behaved. Plug-in estimators of expectation functionals, named V-statistics after von Mises, can be obtained but may be biased. It is, however, always possible to construct an unbiased estimator of expectation functionals regardless of the underlying distribution function $F$ . These estimators are named U-statistics, with the “U” standing for unbiased.

This blog post provides 1) the definitions of symmetric kernels and expectation functionals; 2) an overview of plug-in estimators of expectation functionals or V-statistics; 3) an overview of unbiased estimators for expectation functionals or U-statistics.

Continue reading U-, V-, and Dupree statistics

Plug-in estimators of statistical functionals

Consider a sequence of $n$ independent and identically distributed random variables $X_1, X_2, …, X_n \sim F$ . The distribution function $F$ is unknown but belongs to a known set of distribution functions $\mathcal{F}$ . In parametric estimation, $\mathcal{F}$ may represent a family of distributions specified by a vector of parameters, such as $(\mu, \sigma)$ in the case of the location-scale family. In nonparametric estimation, $\mathcal{F}$ is much more broad and is subject to milder restrictions, such as the existence of moments or continuity. For example, we may define $\mathcal{F}$ as the family of distributions for which the mean exists or all distributions defined on the real line $\mathbb{R}$ .

As mentioned in my previous blog post comparing nonparametric and parametric estimation, a statistical functional is any real-valued function of the cumulative distribution function $F$ , denoted $\theta = T(F)$ . Statistical functionals can be thought of as characteristics of $F$ , and include moments

$T(F) = \mathbb{E}_{F}[X^{k}]$

and quantiles

$T(F) = F^{-1}(p)$

as examples.

An infinite population may be considered as completely determined by its distribution function, and any numerical characteristic of an infinite population with distribution function $F$ that is used in statistics is a [statistical] functional of $F$ .

Wassily Hoeffding. “A Class of Statistics with Asymptotically Normal Distribution.” Ann. Math. Statist. 19 (3) 293 – 325, September, 1948.

This blog post aims to provide insight into estimators of statistical functionals based on a sample of $n$ independent and identically random variables, known as plug-in estimators or empirical functionals.

Continue reading Plug-in estimators of statistical functionals