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Complete Statistic: Definition

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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 completeness 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(Y) has a pdf or pmf f(t|θ). Then T(Y) is a complete statistic if Eθ[g(T(Y))] = 0 for all θ ∈ Θ implies that Pθ[g(T(Y)) = 0] = 1 for all θΘ (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.

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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