Statistical Functionals

Definition (Statistical Functional)

A statistical functional is any function of , i.e., you input a distribution (CDF) and it returns a number.

Definition (Plug-in Estimator)

The plug-in estimator of is defined by

In other words, just plug in for the unknown .

Definition (Linear Functional)

If

for some function , then is called a linear functional.

The reason is called a linear functional is because satisfies

hence is linear in its arguments.

Theorem

The plug-in estimator for linear functional is

Definition (Normal-based Interval)

In many cases, it turns out that

By normal-based confidence intervals, an approximate confidence interval for is then

We call this the Normal-based interval.

Mean of a Distribution

The mean of a distribution is a statistical functional

When has a PDF , . The plug-in estimator is

The standard error is . If denotes an estimate of , then the estimated standard error is . A Normal-based confidence interval for is .

Variance of a Distribution

The variance of a distribution is a statistical functional

When has a PDF , . The plug-in estimator is

Another reasonable estimator of is the sample variance

In practice, there is little difference between and .

Median of a Distribution

Using the concept of a statistical functional, median and any quantile can be easily defined. The median of a distribution is a point such that . Thus,

Sources

• Wasserman, L. (2010). All of Statistics: A concise Course in Statistical Inference. Chapter 7.2.