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.
![\[ \hat{\mu} = \mathbb{E}_{\hat{F}_n}[X] = \sum_{i=1}^{n} X_i P(X = X_i) = \frac{1}{n} \sum_{i=1}^{n} X_i = \bar{X}_{n} \]](https://statisticelle.com/wp-content/ql-cache/quicklatex.com-1c4166355261af485d335606b1d858c2_l3.png "Rendered by QuickLaTeX.com")
![\[ \hat{\sigma}^{2} = \mathbb{E}_{\hat{F}_{n}}[(X- \mathbb{E}_{\hat{F}_n}[X])^2] = \mathbb{E}_{\hat{F}_n}[(X - \bar{X}_{n})^2] = \frac{1}{n} \sum_{i=1}^{n} (X_i - \bar{X}_{n})^2. \]](https://statisticelle.com/wp-content/ql-cache/quicklatex.com-1ae64559278ccc7ec2f0dd38d1713032_l3.png "Rendered by QuickLaTeX.com")
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
![\[ T(F) = \mathbb{E}_{F} ~g(X)\]](https://statisticelle.com/wp-content/ql-cache/quicklatex.com-7bfa10261fdefb6ccf03459f369fae39_l3.png "Rendered by QuickLaTeX.com")
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.