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 ensures that the distributions corresponding to different values of the parameters are distinct. In statistics, completeness is a property of a statistic in relation to a model for a set of observed data. In essence, 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 model Pθ parametrized by θ. Say T is statistic; that is, the composition of a measurable function with a random sample X1,...,Xn. The statistic T is said to be complete for the distribution of X if, for every measurable function g,: if  E θ ⁡ ( g ( T ) ) = 0  for all  θ  then  P θ ( g ( T ) = 0 ) = 1  for all  θ . {displaystyle { ext{if }}operatorname {E} _{ heta }(g(T))=0{ ext{ for all }} heta { ext{ then }}mathbf {P} _{ heta }(g(T)=0)=1{ ext{ for all }} heta .} The statistic T 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 Xi 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 E p ( g ( T ) ) = 0 {displaystyle E_{p}(g(T))=0} if and only if: