SELECTING THE BEST ONE OF SEVERAL BINOMIAL POPULATIONS 569 



This resuK can also be obtained by starting with (A16) with 



P[2] = • • • = Pl^.] = p[ij - Cl* = p* ] 



and integrating by parts. It is clear from (A20) that for fixed p[i] (i ^ 2) 

 the Pes is an increasing function of p[i] and is indeed strictly increasing 

 for p[i] in the unit interval. 



The second step is to hold pn] fixed and to decrease the values of 

 P[i] {i ^ 2). This increases the probability of a correct selection by our 

 previous result above. This proves the monotonicity property for the 

 alternative specification. 



Appendix III 



LARGE SAMPLE THEORY ■ — ORIGINAL SPECIFICATION 



For p[i] > p[2] the probability of a correct selection satisfies the in- 

 equalities 



P{Xa, > X,i){i = 2, 3, ••• ,1-)} < Pes 



< P{Xa) ^ X,i,{i = 2,3, ••• ,/c)} (A21) 



unless p[i] — 1 and p[2] = in which case equality signs hold since the 

 three quantities above are all unity. Letting ^[i] = 1 — p[\] , we can 

 write the left member of (A21) as 



P {Zi > . ^*^ = (t = 1, 2, ...,/. - 1)\ 



vpmgm + (pm - rf*)(?[i] + d*) J 



(A22) 

 where 

 7 = X(i) - Z(.-+i) - nd* (■ ^ -i r, I — ^\ 



(A23) 



For the configuration (Al) with d — d* the chance variables Z,- tend to 

 normal chance variables iV(0,l) with zero mean and unit variance as 

 n — > X . We have purposely omitted any continuity correction in (A22) 

 in order to get a better approximation for the smaller values of Ji. 



To derive the least favorable configuration for large n we can restrict 

 our attention to those configurations given by (Al) with d = d*. The 

 quantity pfi] , which minimizes (A22), is obtained by maximizing the 

 expression in (A22) 



Q(p) = p(l -p)^(p- f/*)(l -p + d*) (A24) 



= -2p' + 2(1 + d*)p - d*(l -\- d*) (A25) 



