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In statistics, a parametric model or parametric family or finite-dimensional model is a family of distributions that can be described using a finite number of parameters. These parameters are usually collected together to form a single ''k''-dimensional ''parameter vector'' ''θ'' = (''θ''1, ''θ''2, …, ''θ''''k''). Parametric models are contrasted with the semi-parametric, semi-nonparametric, and non-parametric models, all of which consist of an infinite set of “parameters” for description. The distinction between these four classes is as follows: * in a “''parametric''” model all the parameters are in finite-dimensional parameter spaces; * a model is “''non-parametric''” if all the parameters are in infinite-dimensional parameter spaces; * a “''semi-parametric''” model contains finite-dimensional parameters of interest and infinite-dimensional nuisance parameters; * a “''semi-nonparametric''” model has both finite-dimensional and infinite-dimensional unknown parameters of interest. Some statisticians believe that the concepts “parametric”, “non-parametric”, and “semi-parametric” are ambiguous.〔, ch.7.4〕 It can also be noted that the set of all probability measures has cardinality of continuum, and therefore it is possible to parametrize any model at all by a single number in (0,1) interval. This difficulty can be avoided by considering only “smooth” parametric models. ==Definition== A parametric model is a collection of probability distributions such that each member of this collection, ''Pθ'', is described by a finite-dimensional parameter ''θ''. The set of all allowable values for the parameter is denoted Θ ⊆ R''k'', and the model itself is written as : When the model consists of absolutely continuous distributions, it is often specified in terms of corresponding probability density functions: : The parametric model is called identifiable if the mapping ''θ'' ↦ ''Pθ'' is invertible, that is there are no two different parameter values ''θ''1 and ''θ''2 such that ''P''''θ''1 = ''P''''θ''2. 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「Parametric model」の詳細全文を読む スポンサード リンク
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