## 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 such that

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

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

A random sample can be used to construct the empirical cumulative distribution function (ECDF) ,

which assigns mass to each .

is a valid, discrete CDF which can be substituted for to obtain . 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 with respect to such that for ,

When is a function symmetric in its arguments such that, for e.g. , it is referred to as a symmetric kernel of degree . If is not symmetric, a symmetric equivalent can always be found,

where represents the set of all permutations of the indices .

A statistical functional belongs to a special family of expectation functionals when:

1. , and
2. is a symmetric kernel of degree .

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

so that is the average of evaluated at all possible permutations of size from . Since the can appear more than once within each summand, 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 is large enough, it can be shown that a U-statistic can always be constructed for expectation functionals.

Since is symmetric, we can require that , resulting in combinations of the subscripts . The U-statistic is then the average of evaluated at all distinct combinations of ,

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

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-, 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, , and explained how plug-in estimators could be constructed by substituting the empirical cumulative distribution function (ECDF) for the unknown CDF . Plug-in estimators of the mean and variance were provided and used to demonstrate plug-in estimators’ potential to be biased.

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

1) is the expectation of a function with respect to the distribution function ; and

2) the function 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 . 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.

## Parametric vs. Nonparametric Approach to Estimations

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

To demonstrate the parametric approach, consider independent and identically distributed random variables generated from an exponential distribution with rate . Investigators wish to estimate the 75 percentile and erroneously assume that their data is normally distributed. Thus, is assumed to be the Normal CDF but and are unknown. The parameters and are estimated in their typical way by and , respectively. Since the normal distribution belongs to the location-scale family, an estimate of the percentile is provided by,

where 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 percentile of is 0.69 while the parametric estimate is 0.88.

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

Any statistic that can be expressed as a function of the CDF, known as a statistical functional and denoted , can be estimated by substituting for . That is, plug-in estimators can be obtained as .

## Motivation

For observed pairs , , the relationship between and can be defined generally as

where and . If we are unsure about the form of , our objective may be to estimate without making too many assumptions about its shape. In other words, we aim to “let the data speak for itself”.

Simulated scatterplot of . Here, and . The true function is displayed in green.

Non-parametric approaches require only that be smooth and continuous. These assumptions are far less restrictive than alternative parametric approaches, thereby increasing the number of potential fits and providing additional flexibility. This makes non-parametric models particularly appealing when prior knowledge about ‘s functional form is limited.

## Estimating the Regression Function

If multiple values of were observed at each , could be estimated by averaging the value of the response at each . However, since is often continuous, it can take on a wide range of values making this quite rare. Instead, a neighbourhood of is considered.

Result of averaging at each . The fit is extremely rough due to gaps in and low frequency at each .

Define the neighbourhood around as for some bandwidth . Then, a simple non-parametric estimate of can be constructed as average of the ‘s corresponding to the within this neighbourhood. That is,

(1)

where

is the uniform kernel. This estimator, referred to as the Nadaraya-Watson estimator, can be generalized to any kernel function (see my previous blog bost). It is, however, convention to use kernel functions of degree (e.g. the Gaussian and Epanechnikov kernels).

The red line is the result of estimating with a Gaussian kernel and arbitrarily selected bandwidth of . The green line represents the true function .