distribution families Archives • Statisticelle

Consider a sequence of $n$ independent and identically distributed random variables $X_1, X_2, …, X_n \sim F$ . The distribution function $F$ is unknown but belongs to a known set of distribution functions $\mathcal{F}$ . In parametric estimation, $\mathcal{F}$ may represent a family of distributions specified by a vector of parameters, such as $(\mu, \sigma)$ in the case of the location-scale family. In nonparametric estimation, $\mathcal{F}$ is much more broad and is subject to milder restrictions, such as the existence of moments or continuity. For example, we may define $\mathcal{F}$ as the family of distributions for which the mean exists or all distributions defined on the real line $\mathbb{R}$ .

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 $F$ , denoted $\theta = T(F)$ . Statistical functionals can be thought of as characteristics of $F$ , and include moments

$T(F) = \mathbb{E}_{F}[X^{k}]$

and quantiles

$T(F) = F^{-1}(p)$

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

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

Continue reading Plug-in estimators of statistical functionals