Abstract
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.
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References
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Basu, D. (2011). On Statistics Independent of a Complete Sufficient Statistic. In: DasGupta, A. (eds) Selected Works of Debabrata Basu. Selected Works in Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-5825-9_14
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DOI: https://doi.org/10.1007/978-1-4419-5825-9_14
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