Completeness (statistics)

In statistics, completeness is a property of a statistic in relation to a model for a set of observed data. In essence, it is a condition which ensures that the parameters of the probability distribution representing the model can all be estimated on the basis of the statistic: it ensures that the distributions corresponding to different values of the parameters are distinct.

It is closely related to the idea of identifiability, but in statistical theory it is often found as a condition imposed on a sufficient statistic from which certain optimality results are derived.

Consider a random variable X whose probability distribution belongs to a parametric family of probability distributions P_θ parametrized by θ.

Formally, a statistic s is a measurable function of X; thus, a statistic s is evaluated on a random variable X, taking the value s(X), which is itself a random variable. A given realization of the random variable X(ω) is a data-point (datum), on which the statistic s takes the value s(X(ω)).

The statistic s is said to be complete for the distribution of X if, for every measurable function g,

The statistic s is said to be boundedly complete for the distribution of X if this implication holds for every measurable function g that is also bounded.

The Bernoulli model admits a complete statistic. Let X be a random sample of size n such that each X_i has the same Bernoulli distribution with parameter p. Let T be the number of 1s observed in the sample. T is a statistic of X which has a binomial distribution with parameters (n,p). If the parameter space for p is (0,1), then T is a complete statistic. To see this, note that

Observe also that neither p nor 1 − p can be 0. Hence ${\displaystyle E(g(T))=0}$ if and only if:

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