## Mathematical Statistics Lesson of the Day – Basu’s Theorem

Today’s Statistics Lesson of the Day will discuss Basu’s theorem, which connects the previously discussed concepts of minimally sufficient statistics, complete statistics and ancillary statistics.  As before, I will begin with the following set-up.

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) or probability mass function (PMF) for $X_1, X_2, ..., X_n$.

Let

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

be a statistics based on $\textbf{X}$.

Basu’s theorem states that, if $T(\textbf{X})$ is a complete and minimal sufficient statistic, then $T(\textbf{X})$ is independent of every ancillary statistic.

Establishing the independence between 2 random variables can be very difficult if their joint distribution is hard to obtain.  This theorem allows the independence between minimally sufficient statistic and every ancillary statistic to be established without their joint distribution – and this is the great utility of Basu’s theorem.

However, establishing that a statistic is complete can be a difficult task.  In a later lesson, I will discuss another theorem that will make this task easier for certain cases.

## Mathematical Statistics Lesson of the Day – An Example of An Ancillary Statistic

Consider 2 random variables, $X_1$ and $X_2$, from the normal distribution $\text{Normal}(\mu, \sigma^2)$, where $\mu$ is unknown.  Then the statistic

$D = X_1 - X_2$

has the distribution

$\text{Normal}(0, 2\sigma^2)$.

The distribution of $D$ does not depend on $\mu$, so $D$ is an ancillary statistic for $\mu$.

Note that, if $\sigma^2$ is unknown, then $D$ is not ancillary for $\sigma^2$.

## Mathematical Statistics Lesson of the Day – Ancillary Statistics

The set-up for today’s post mirrors my earlier Statistics Lessons of the Day on sufficient statistics and complete 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) or probability mass function (PMF) for $X_1, X_2, ..., X_n$.

Let

$a = A(\mathbf{X})$

be a statistics based on $\textbf{X}$.

If the distribution of $A(\textbf{X})$ does NOT depend on $\theta$, then $A(\textbf{X})$ is called an ancillary statistic.

An ancillary statistic contains no information about $\theta$; its distribution is fixed and known without any relation to $\theta$.  Why, then, would we care about $A(\textbf{X})$  I will address this question in later Statistics Lessons of the Day, and I will connect ancillary statistics to sufficient statistics, minimally sufficient statistics and complete statistics.