Completeness (statistics)

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Completeness in statistics is a property of a statistic T computed from a sample X1, ..., Xn drawn from a parametric model Pθ. T is complete for the model if the only function g of T with expected value Eθ[g(T)] equal to zero for every θ is the zero function (g ≡ 0 almost everywhere). In other words, if a nonzero function of T always has a nonzero average, then T is complete. A related idea is bounded completeness, which requires this to hold for all bounded functions g.

Completeness is closely related to the idea of sufficiency. A statistic that is both complete and sufficient has especially strong usefulness: by the Lehmann–Scheffé theorem, it yields the best unbiased estimator of the parameter θ based on the data.

Examples:
- Bernoulli model: Take n independent Bernoulli(p) trials and let T be the number of successes. Then T ~ Binomial(n, p). If p ∈ (0,1), T is a complete statistic for p. Intuitively, if a function g(T) has zero expectation for every p in (0,1), then g must be zero for all possible T.
- Normal model with known variance: If X1 and X2 are independent N(θ, 1), the sum X1 + X2 is a complete and sufficient statistic for θ. The vector (X1, X2) is sufficient but not complete (for example, X1 − X2 is an unbiased estimator of zero).

Notes:
- Many models have sufficient statistics that are not complete, so the Lehmann–Scheffé theorem cannot be applied in those cases.
- Bounded completeness leads to Basu’s theorem: a boundedly complete and sufficient statistic is independent of any ancillary statistic. Bahadur’s theorem is another related result.
- In some models there exists a minimal sufficient statistic that is complete; in others, the minimal sufficient statistic is not complete, so other estimators may be preferable.

A helpful illustration: in a scale-family model like the uniform distribution on [0, θ], the minimum and maximum are jointly sufficient but not complete, showing that completeness is not automatic and must be checked for each model.