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

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.

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) 
## [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)
## [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

Kernel Regression


For observed pairs (x_i, y_i), i = 1, …, n, the relationship between x and y can be defined generally as

    \[ y_i = m(x_i) + \varepsilon_i \]

where f(x_i) = E[y_i | x = x_i] and \varepsilon_i \stackrel{iid}{\sim} (0, \sigma^2). If we are unsure about the form of m(\cdot), our objective may be to estimate m(\cdot) without making too many assumptions about its shape. In other words, we aim to “let the data speak for itself”.

Simulated scatterplot of y = f(x) + \epsilon. Here, x \sim Uniform(0, 10) and \epsilon \sim N(0, 1). The true function f(x) = sin(x) is displayed in green.

Non-parametric approaches require only that m(\cdot) 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 m(\cdot)‘s functional form is limited.

Estimating the Regression Function

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

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

Define the neighbourhood around x as x \pm \lambda for some bandwidth \lambda > 0. Then, a simple non-parametric estimate of m(x) can be constructed as average of the y_i‘s corresponding to the x_i within this neighbourhood. That is,

(1)   \begin{equation*} \hat{f}_{\lambda}(x) = \frac{\sum_{n} \mathbb{I}(|x - x_i| \leq \lambda)~ y_i}{\sum_{n} \mathbb{I}(|x - x_i| \leq \lambda)} = \frac{\sum_n K\left( \frac{x - x_i}{\lambda} \right) y_i}{\sum_n K\left( \frac{x - x_i}{\lambda} \right) } \end{equation*}


    \[ K(u) = \begin{cases} \frac{1}{2} & |u| \leq 1 \\ 0 & \text{o.w.} \end{cases} \]

is the uniform kernel. This estimator, referred to as the Nadaraya-Watson estimator, can be generalized to any kernel function K(u) (see my previous blog bost). It is, however, convention to use kernel functions of degree \nu = 2 (e.g. the Gaussian and Epanechnikov kernels).

The red line is the result of estimating f(x) with a Gaussian kernel and arbitrarily selected bandwidth of \lambda = 1.25. The green line represents the true function sin(x).

Continue reading Kernel Regression