On Statistics Independent of a Complete Sufficient Statistic

By D. Basu. 2011

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If {Ρ θ}, θЄΩ, be a family of probability measures on an abstract sample space \(\mathcal {G}\) and Τ be a sufficient statistic for θ then for a statistic T 1 to be stochastically independent of Τ it is necessary that the probability distribution of T 1 be independent of θ. The condition is also sufficient if Τ be a boundedly complete sufficient statistic. Certain well-known results of distribution theory follow immediately from the above considerations. For instance, if x 1, x 2,. . . , x n , are independent Ν(μ, σ)’s then the sample mean \(\bar x\) and the sample variance s 2 are mutually independent and are jointly independent of any statistic f (real or vector valued) that is independent of change of scale and origin. It is also deduced that if x 1, x 2, . . ., x n are independent random variables such that their joint distribution involves an unknown location parameter θ then there can exist a linear boundedly complete sufficient statistic for θ only if the x’s are all normal. Similar characterizations for the Gamma distribution also are indicated.

Subfields: Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/3d1ab78231dbca7b3b63443213fc432176fe5ede

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