ePositive
2000
We do not know wlrich units are positive or how many are positive unless z -- l. This positive/negntive t,erminology gives us il wider scope in applications since the positivc units r m y I-)c thc oncs we would rather not find or they may be thc orrcs we arc: trying to find. Tlie problcrri in this paper is to firrtl n single positivc wiit, assi~mirig N -- (XI, so tlmt (with p > 0) the prok)ahility is urrit,y th;it it exis1,s. For the two settings we consider: (1) q is ltriow~r and (2) q is given I)y the Uniforrn(0,l) prior, the first one ha.s alreatly appeared ill the literature, Irmt not the secorid. We also wish to define a very simple (nonyoptimitl) procctl~m (called DH for tloublirrg a11tl halving), and sec how it cornpares with the optimal procedure in each of these two settings. Tllc efficiency of a procedure is defined by conlparing it,s expected r1rnnbc.r of tests rcquirctl to that of the opti1na.l procedure. &s;I&s;hc D11 proccdurc does quite wcll in the first setting, rcaching 94.22% efficiency (for some q values), but in the secontl setting, it tlocs much hctter with a (B;~,yesian) efficiency of 99.6%. This is a rernar1i;hlc result for a proccclure so easy to ii-nplcrnc~llt.