Profile Sufficiency
DOI:
https://doi.org/10.17713/ajs.v35i2&3.360Abstract
Let P = {P?1,?2 ; (?1; ?2) ? ?1 × ?2} be a family of probability measures on a measurable space (X;A) parameterized by a pair of abstract valued parameters ?1; ?2. A statistic T1 is called profile sufficient for ?1 if for any fixed ?2 ? ?2, T1 is sufficient for ?1.
For a dominated family P, a necessary and sufficient condition in the form of a factorization theorem is proved for T1 to be profile sufficient for ?1 and for a statistic T2 to be profile sufficient for ?2. The classical (Halmos - Savage) factorization theorem is its special case corresponding to T1 = T2.
If Ti is profile sufficient for ?i ; i = 1; 2 and a statistic S is independent of T1 and (separately) of T2 (but S is not assumed independent of (T1; T2)) for all ?1; ?2, then S is ancillary.
References
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