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.