A **complete statistic ****T** “… is a complete statistic if the family of probability densities {g(t; θ) is complete” (Voinov & Nikulin, 1996, p. 51).

The concept is perhaps best understood in terms of the Lehmann-ScheffĂ© theorem “…if a sufficient statistic is boundedly complete it is minimal sufficient. The converse is false” (Cox & Hinkley, p. 31). **Boundledly complete** means that you have no uninformative mean values (i.e. those not dependent on T). In other words, no non-trivial function of **T** has constant mean value.

## The Problem with a Complete Statistic

Many definitions in statistic are intuitive, but unfortunately “complete statistic” is not one of them. Take the first definition given above. This poses a couple of (perhaps obvious) problems:

- The definition includes the word “complete”,
- The definition doesn’t give any information for determining whether or not any particular statistic is complete.

The definition of a complete statistic then is somewhat misleading and should more accurately reflect the fact that c**ompleteness indicates that a family of distributions, for all possible values of θ, provides a sufficiently rich set of vectors ** (Cremling, n.d.).

## Formal Definition of Complete Statistic

A **complete statistic** is formally defined as:

Suppose a statistic

(T) has a pdf or pmfYf(|t). Thenθ(T) is a complete statistic if EY_{θ}[g((T))] = 0 for all θ ∈ Θ implies that PY_{θ[g(T(Y)) = 0] = 1 for all θ ∈ Θ (Olive, 2014).}

## Complete Sufficient Statistic

It’s possible for a complete statistic to provide no information at all about θ. In order for complete statistics to be useful, they must also be a sufficient statistic; A sufficient statistic summarizes all of the information in a sample about a chosen parameter. Ideally then, a statistic should ideally be complete and sufficient, which means that:

- The statistic isn’t missing any information about θ and
- Doesn’t provide any irrelevant information (Shynk, 2012).

Specifically, **a complete statistic is one that is minimal sufficient**.

**Related article:** Complete Sequence (CalculusHowTo.com).

## References

Cox, D. & Hinkley, D. (1979). Theoretical Statistics 1st Edition. Chapman and Hall/CRC.

Cremling, D. Completeness and Sufficiency. Retrieved May 19, 2020 from: https://math.ou.edu/~cremling/teaching/lecturenotes/stat/ln5.pdf

Olive, D. (2014). Statistical Theory and Inference. Springer.

Shynk, J. (2012). Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications. Wiley.

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