Mathematical Statistics Lesson of the Day – Ancillary Statistics

June 24, 2015

(This article was originally published at The Chemical Statistician » Statistics, and syndicated at StatsBlogs.)

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.