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.

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.

Mathematical Statistics Lesson of the Day – Sufficient Statistics

*Update on 2014-11-06: Thanks to Christian Robert’s comment, I have removed the sample median as an example of a sufficient statistic.

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}.  Let g_\theta(t) be the PDF for T(X).

If the conditional PDF

h_\theta(\mathbf{X}) = f_\theta(x) \div g_\theta[T(\mathbf{X})]

is independent of \theta, then T(\mathbf{X}) is a sufficient statistic for \theta.  In other words,

h_\theta(\mathbf{X}) = h(\mathbf{X}),

and \theta does not appear in h(\mathbf{X}).

Intuitively, this means that T(\mathbf{X}) contains everything you need to estimate \theta, so knowing T(\mathbf{X}) (i.e. conditioning f_\theta(x) on T(\mathbf{X})) is sufficient for estimating \theta.

Often, the sufficient statistic for \theta is a summary statistic of X_1, X_2, ..., X_n, such as their

  • sample mean
  • sample median – removed thanks to comment by Christian Robert (Xi’an)
  • sample minimum
  • sample maximum

If such a summary statistic is sufficient for \theta, then knowing this one statistic is just as useful as knowing all n data for estimating \theta.

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