The term “ancillary statistic” is one of those terms that mean something slightly different depending on where you read about it.

**Most (but not all) authors agree** that an ancillary statistic is a distribution constant statistic that can be combined with a maximum likelihood estimator to create a minimally sufficient statistic. The ancillary statistic in this sense is defined as **one part of a sufficient statistic** that has a parameter free marginal distribution.

**Some authors** will describe an ancillary statistic as simply a statistic whose distribution doesn’t depend on the model parameters. In this context, an ancillary statistic is basically a summary of the data. Standing alone, it doesn’t give any information about the parameter P. For example, an ancillary statistic could be an estimator for a random sample size, but it isn’t an estimator for any *specific *sample size (Kardaun, 2006).

## As a Complement to Sufficient Statistics

Given the above definitions, it should be easy to see why ancillaries are sometimes referred to as the complement of a sufficient statistic. While a sufficient statistic contains *all *of the parameter’s information, an ancillary contains **no** information about the model parameter. Basu’s Theroem (as cited in Gosh, 2011) summarizes this idea:

If U is a complete statistic

anda sufficient statistic for a parameter(θ),andif V is an ancillary statistic for θ, then U and V are independent.

Ancillary statistics are used in compound distributions and unconditional inference.

## Specific Types of Ancillary Statistic

**First order ancillary:**an ancillary is first order if the statistic’s expected value is independent of the population.**Trivial ancillary:**Defined as the constant statistic V(X) ≡ c ∈ ℝ (Shao, 2008).

## Advantages

If an ancillary contains no information about population parameters, why use it at all? Ning-Zhing & Jian (2008) state two reasons why you might choose ancillary over sufficiency:

- Invariance to the parameter.
**Invariant statistics**are not easily changed by transformations, like simple data shifts. - Independence to sufficient statistics.

## References

Ghosh M. (2011) Basu’s Theorem. In: DasGupta A. (eds) Selected Works of Debabrata Basu. Selected Works in Probability and Statistics. Springer, New York, NY

Kardaun (2006). Classical Methods of Statistics: With Applications in Fusion-Oriented Plasma Physics. Springer Science & Business Media.

Ning-Zhing, S. & Jian, T. (2008). Statistical Hypothesis Testing: Theory and Methods. World Scientific.

Shao, J. Mathematical Statistics. Springer Science & Business Media.

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