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.


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:

  1. T(F) = \mathbb{E}_F ~\phi(X_1, …, X_a), and
  2. \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