Asymptotic efficiency of a sequential multihypothesis test

that leading to (7) Theorem 1 implies REFERENCES 1 ; log $ 5 ; log-l-a! + ; " , " p log l/P n [ (+%y-(x " ,Y ") < l%M log cy-+-n n I which holds for all sufficiently large n because of (13). Then, (14) follows from (12) and the above inequality. 0 We conjecture that the bound (14) is in fact tight; however, the known approaches to the constructive part of the coding theorem are not sufficient to prove this conjecture even for the simplest channels (for which the reliability function is not yet known for all rates). For example, in the case of a binary-symmetric channel, the evaluation of the right-hand side of (14) is an interesting unsolved large-deviations/optimization problem. Theorem 1 admits a very simple proof that is quite different from the proofs of the special cases in [2] and [4]. placed in decreasing order, pointwise in the sample space' (it is immaterial how ties are resolved). First note from (3) that E = 1-E{Zi}. (15) For any o E [O,l], we can write P(,(XIY) > a) = E { c rr(L]Y)l{n(LIY) > CX} (16) &X 1 where the expectation is with respect to the unconditional distribution of Y. The argument of the expected value in (16) can be written as c ~(klY)l{~(klY) > a} = c zkl{zk > a}. (17) kEX &X Dropping all but the first term P(7r(XIY) > a) 2 E{&l{Z1 > CX}}. (18) In view of (15) and (18), all we need to do is to relate E{& } to E{Zl l{Zi > e}} using the fact that 0 5 21 5 1. Since 21 5 1 note that, for any o E [0, l] we have 2' = a& + (1-a)& i: Q + (1-a)Z11{21 > G} (19) which is tantamount to upperbounding 21 by cy + (1-cr)Zi when o < 21 5 1, and by cy, otherwise. Thus on combining (18) and (19), we have E{ZI} I cv + (1-cu)E{&l{Z~ > a>> 5 a+(l-cy)P(7r(XIY) > a) (20) which, together with (15) implies the bound. cl 'That is, for each point w in the underlying sample space, Z,(w), Z,(w),..., denotes theordered sequence n(llY(w)),~(llY(w)),.... Abstract-A sequential multihypothesis test known as the MSPRT is generalized to account for nonuniform decision costs. Bounds on error probabilities and asymptotic expressions for the stopping time and error probabilities are given. A key result of this …