## Mathematical Statistics Lesson of the Day – Complete Statistics

The set-up for today’s post mirrors my earlier Statistics Lesson of the Day on sufficient statistics.

Suppose that you collected data

$\mathbf{X} = X_1, X_2, ..., X_n$

in order to estimate a parameter $\theta$.  Let $f_\theta(x)$ be the probability density function (PDF)* for $X_1, X_2, ..., X_n$.

Let

$t = T(\mathbf{X})$

be a statistic based on $\mathbf{X}$.

If

$E_\theta \{g[T(\mathbf{X})]\} = 0, \ \ \forall \ \theta,$

implies that

$P \{g[T(\mathbf{X})]\} = 0] = 1,$

then $T(\mathbf{X})$ is said to be complete.  To deconstruct this esoteric mathematical statement,

1. let $g(t)$ be a measurable function
2. if you want to use $g[T(\mathbf{X})]$ to form an unbiased estimator of the zero function,
3. and if the only such function is almost surely equal to the zero function,
4. then $T(\mathbf{X})$ is a complete statistic.

I will discuss the intuition behind this bizarre definition in a later Statistics Lesson of the Day.

*This above definition holds for discrete and continuous random variables.