Plug-in estimators of statistical functionals

Consider a sequence of independent and identically distributed random variables . The distribution function is unknown but belongs to a known set of distribution functions . In parametric estimation, may represent a family of distributions specified by a vector of parameters, such as in the case of the location-scale family. In nonparametric estimation, is much more broad and is subject to milder restrictions, such as the existence of moments or continuity. For example, we may define as the family of distributions for which the mean exists or all distributions defined on the real line .

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 , denoted . Statistical functionals can be thought of as characteristics of , and include moments

and quantiles

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 that is used in statistics is a [statistical] functional of .

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 independent and identically random variables, known as plug-in estimators or empirical functionals.