Estimators

01An estimator is a rule that turns a sample into a guess about a population quantity#

An estimator is a rule that turns a sample into a guess about a population quantity. The rule exists before collecting data; an estimate is its particular numerical output afterward. Repeated samples can give different estimates even when the underlying population stays the same.

An estimator is a rule that maps sample data to a number used to estimate a population quantity. Before the data arrive, its output can vary. An estimate is the number obtained from the observed data.

Take the supplied rule $T(x,y)=x+y$. Possible data $(1,2)$ give $3$; data $(2,4)$ give $6$. The rule stays the same.

For a rule that averages two observations, the sample $(2,6)$ gives estimate 4 while $(3,9)$ gives 6. These are two outputs of the same estimator. Changing from a mean to a maximum would instead change the estimation rule and generally the population quantity it is suited to estimate.

For the supplied rule $T(x,y)=(x+y)/2$, add the two inputs and divide by $2$. With observed inputs $2,6$, the estimate is $4$. It is fixed for those data.

02A fresh random sample can produce a new estimate#

A fresh random sample can produce a new estimate. Re-running a deterministic rule on the same saved data produces the same value. The unknown population target does not become equal to the estimate by definition.

03For T(x,y)=max(x,y), choose the larger input#

For $T(x,y)=\max(x,y)$, choose the larger input. For $T(x,y)=x$, choose the first. Any supplied deterministic rule has the same distinction: rule before evaluation, fixed number afterward.

04A closer look#

When comparing ways to estimate a model’s average error, ask both what population quantity is targeted and how the rule varies across samples. A single close estimate does not establish that the method is unbiased or reliably accurate.