568 THE BELL SYSTEM TECHNICAL JOURNAL, >L\.RCH 1957 



strictly increasing, i.e., for fixed p^j the Pes is a strictly increasing func- 

 tion of each of the differences pm — p[i] (i ^ 2) as was to be shown. 



It follows from the above result that in searching for a least favorable 

 configuration among all those in which the experimenter wants his 

 specification satisfied w^e may restrict our attention to those of the form 

 (Al). Moreover, we may set d in (Al) equal to d* since, for d > d* and 

 fixed p[i] , the difference d — d* may be added to each p[i] {i ^ 2) and 

 the probability of a correct selection is increased. Then (A 15) reduces to 



/n+l/2 

 G'-\y;Pw - d*)g(r,Pii,) dy (A16) 



-1/2 



It was shown in the section on the least favorable configuration that 

 there is a value p[i\ of p{\] which when substituted in (A16) gives the 

 minimum value Pes oi Pes • 



We can now prove the following result in which p[i] is not fixed. For 

 any specified pair p*2] ^ p*i] the probability Pes of a correct selection 

 is smaller for the configuration 



P[i] = P*i] ; P*2] = Pm = Pm = •■' = p[k] (A17) 



than for any configuration given by 



P[i] ^ pti] ; pfo] ^ Pm ^ P[3] ^ ••• ^ Pik] (A18) 



This is shown by considering two separate steps. 



The first step is to increase p[i] holding all the other p's fixed at p[2] . 

 For any arbitrary set of values of pn] with pn] > pi2] the probability 

 of a correct selection can be written as 



Pes =J: f"^''\ n G{y,p,,,)'] [1 - G(y,pn,)]g(y,pu^) dy (A19) 



;=2 J-ll2 L»=2. if^i J 



by adding the probabihties that 



Y(i) > yu) > niin { F(2) , • • • , Yu-d , Yu+d , " , Y^k) } 



for 



For 



j = 2,3,---,k 



Pm > Pm = Pm = • ■ • = Pm = ph 

 this reduces to 



Pes = (k _ 1) r^ " [1 - G(y;p,^,)]G'-'(y;pt2d9(y,ph) dy (A20) 



J-1I2 



