Summary
A recent paper byKumar/Pathak [1977] contains two examples showing how—in the presence of a boundedly complete sufficient statistic—a randomization kernel can be obtained which yields a random variable equivalent to the original one.
It is the purpose of this note to present a simple general theorem containing these examples as special cases. The essential assumption is that the observation can be expressed by a pair of stochastically independent statistics, one of which is ancillary.
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References
Basu, D.: On statistics independent of a complete sufficient statistic. Sankhyā15, 1955, 377–380.
Berk, R.H.: A note on sufficiency and invariance. Ann. Math. Stat.43, 1972, 647–650.
Halmos, P.R.: Measure Theory. Princeton 1950.
Kumar, A., andP.K. Pathak: Two applications of Basu's Lemma. Scand. J. Statist.4, 1977, 37–38.
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Pfanzagl, J. A special representation of a sufficient randomization kernel. Metrika 28, 79–81 (1981). https://doi.org/10.1007/BF01902880
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DOI: https://doi.org/10.1007/BF01902880