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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. ==Definition== 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'' (which must be independent of ''θ'') the following implication holds:〔Young, G. A. and Smith, R. L. (2005). Essentials of Statistical Inference. (p. 94). Cambridge University Press.〕 :E(''g''(''s''(''X''))) = 0 for all ''θ'' implies that ''P''θ(''g''(''s''(''X'')) = 0) = 1 for all ''θ''. The statistic ''s'' is said to be boundedly complete if the implication holds for all bounded functions ''g''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Completeness (statistics)」の詳細全文を読む スポンサード リンク
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