By Hideo Kusuoka and Julien I.E. Hoffman
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If the particular case just considered is modified slightly, so that r has 31 elements and the corresponding matrix is 123 245 3 5 6 then there are two minimal acceptability bases, namely and This modified case is interesting because the frequencies with which ml' m2, and m3 occur, namely 6, 11 and 14 respectively, are the same whether it is H1 or H2 which happens to be the case. If this particular problem was formulated in terms of the simple probability model one would be 'distinguishing' between two identical probability distributions!
N = = r 1 x r 2 x ... x r n and the corresponding situation schemas are [/12 ... n,i = (r 12 ... n> C12 •.. n,;. R 12 ... n), i = 1,2 39 THE POVERTY OF STATISTICISM where R12 ... " = Rl X R z x ... X R" and Let M 12 ... ,. be Ml xMz x··· xM,. and let P12 ... ,I: r 12 ... ,. --+ M12 ... ,. be defined by P12 ... ) = (Pl,bl), Pz,bz), ... », so that for (mlo mz, ... ) in M12 ... ,. 1) N(Plf ... ,I(ml,mz, ... » = ,. n N(p;} (m ». t t=1 If we observe mlo mz, ... , mIl in the experiments Clo Cz, ... , Cn respectively then the acceptabilities for the hypotheses H1 and Hz from the combined experiment are given by I A12 ...
K) g~'g;2 ... g~k where the summation is over all non-negative integers r1> r2, ... , rk with r1 +r2+···+ rk=n. 3). 6). Write and let S6 be the set of k-tuples of non-negative integers (rl' r2, ... , rk) with r1 +r2+ ... +rk=n and I~ - g11 > ~ 44 P. D. FINCH for at least onej=l, 2, ... , k. 7) holds. 4), b 1 (ep~'p~Z ... p~k) ~ Min [e (1 - 8)",1] whenever (rl' r2' ... 8) o ~ b,,+l (e) ~ L Q (rl' r2, ... , rk) + Min [e(l - 8)",1]. 6) the first term tends to zero as n tends to infinity and, quite obviously the same is true of the second term.