March 22 2003 2 3 Minimal su ciency and the Lehmann Sche e property If a statistic T for example a real valued statistic is su cient for a family P of laws then for any other statistic U say with values in Rk the statistic T U with values in Rk 1 is also su cient In terms of algebras if the family P is de ned on a algebra B and a sub algebra A is su cient for P then any other algebra C with A C B is also su cient But since the idea of su ciency is data reduction one would like to have a su cient algebra as small as possible or a su cient statistic of dimension as small as possible A algebra A will be called minimal su cient for P if it is su cient and for any su cient algebra C and each A A there is a C C such that 1A 1C a s for each P P So A is included in C up to almost sure equality of sets Then a statistic T with values in a measurable space Y F will be called minimal su cient i T 1 F is a minimal su cient algebra Example Let P be a family of symmetric laws on R such as the set of all normal laws N 0 2 0 Considering n 1 for simplicity the identity function x is always a su cient statistic but it is not minimal su cient in this case where x is also su cient For dominated families a minimal su cient algebra always exists 2 3 1 Theorem Bahadur Let P be a family of laws on a measurable space S B dominated by a nite measure Then there is always a minimal su cient algebra A for P Also there is such a algebra A containing all sets B in B for which P B 0 for all P P and such an A is unique Proof Take a law equivalent to P from Lemma 2 1 6 d Choose densities dP d for all P P and let A be the smallest algebra for which all the dP d are measurable Then by Theorem 2 1 4 A is su cient Next let C be any su cient algebra for P Let A1 be the collection of sets A in A for which there exists a C C with 1C 1A a s for every P P Then A1 is a algebra since if 1A 1C a s for all P P the same is true for the complements with 1 1A 1 1C and if 1A j 1C j a s for all P the same is true for the union of the sequences A j and C j By the proof of Theorem 2 1 4 c implies b each dP d must equal a C measurable function a s Thus the sets dP d t for each P P and real number t are in A1 Since these sets generate A RAP Theorem 4 1 6 A1 A and A is minimal su cient By choice of the collection Z of sets B in B with P B 0 for all P P is the same as B B B 0 The algebra Y generated by Z and A is easily seen to be minimal su cient If we start with any other minimal su cient algebra C in place of A it follows easily from the minimal su ciency of both A and C that the resulting Y will be the same So Y is uniquely determined The algebra Y just treated may be called the minimal su cient algebra although as a collection of sets it is actually the largest of all minimal su cient algebras An idea closely related to minimal su ciency is the Lehmann Sche e property as follows 1 De nition Given a collection P of laws on a A B will be called a Lehmann Sche e LS measurable function with f dP 0 for all P A statistic will be called an LS statistic for P measurable is LS for P measurable space S B a sub algebra algebra for P i whenever f is an A P we have f 0 a s for all P P i the smallest algebra for which it is Lehmann and Sche e called algebras satisfying their property complete This is di erent from the notion of complete class of decision rules Also in measure theory a algebra S may be called complete for a measure if it contains all subsets of sets of measure 0 The Lehmann Sche e property is evidently quite di erent So it seemed appropriate to name it here after its authors It is equivalent to uniqueness of A measurable unbiased estimators 2 3 2 Theorem A sub algebra A is LS for P if and only if for every real valued function g on P having an unbiased A measurable estimator the estimator is unique up to equality a s for all P P Proof The constant function 0 always trivially has an unbiased estimator by the statistic which is identically 0 and so measurable for any A Uniqueness of this estimator up to equality a s for all P P yields the de nition of the LS property Conversely if A is LS for P suppose T and U are both A measurable and both unbiased estimators of a function g on P Then T U has integral 0 for all P P so T U 0 a s and T U a s for all P P Some algebras are LS just because they are small For example the trivial algebra S is always LS For any measurable set A the algebra A Ac S is LS for P unless P A is the same for all P in P So LS algebras will be interesting only when they are large enough One useful measure of being large enough is su ciency If a function g on P has an unbiased estimator U and A is a su cient algebra then T EP U A which doesn t depend on P P by Theorem 2 1 8 is an unbiased A measurable estimator as in Corollary 2 2 3 and Theorem 2 3 2 From here on the LS property will be considered for su cient algebras These must be minimal su cient 2 3 3 Theorem For any collection P of laws on a measurable space S B any LS su cient algebra C is minimal su cient Proof If not there is a su cient algebra A and a set C C such that there is no set A in A for which 1C 1A a s for all P P Let f EP 1C A for all P P by Theorem 2 1 8 For some P P f is not equal to 1C a s P otherwise letting A f 1 would give a contradiction We have 1C f f dP 0 as can be seen …