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Statistics, Completeness in
Completeness is a probabilistic property that bestows a form of uniqueness on a statistic h(X) in the context of a statistical model Mθ(x) and crystallizes the link between sufficient and ancillary statistics.
Ronald A. Fisher’s (1922) path-breaking paper recast Karl Pearson’s descriptive statistics (Yule, 1916) into a model-based statistical induction grounded on the concept of a parametric statistical model whose generic form is:
where R := (−∞, ∞), f(x; θ) denotes the (joint) distribution of the sample X := (X1, …, Xn), RXn the sample space, and Θ the parameter space. The likelihood function is defined by: L(θ; x0) ∝ f(x0; θ), θ ∈ Θ.
Example 1. The simple Normal (N) model is specified by:
Example 2. The simple Bernoulli (Ber) model is specified by:
where X = (1/n)k ...
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