SELECTING THE BEST OXE OF SEVERAL BIXOML\L POPULATIONS 575 



where f(iv) and F(w) are defined above, P* and b are known constants 

 with 1/k < P* < 1 and b > and the argument k is a positive integer. 

 Let £ be a (small) fixed number such that < e < Min (P*, 1 — P*). 

 Then c < P* — l/k for sufficiently large A;. Let A = A{e) be defined by 



f /(ly) Jit; = 1 - £ (A54) 



so that 



< f F''~\iD + ?> aA)/(mO rfw ^ £ (A55) 



•'|to|>.l 



for any integer k ^ 1, any n > Q and any 6 > 0. Let n' and n" be the 

 unique positive decimal solutions, respectively, of the equations 



1 F'"'(w + b \/n')I{w) dw = P* - £ (A56) 



f F'"\w + ^ \//^)/(«^) dw = P* (A57) 



where P*, b and ^- are the same as in (A53). It follows from (A55), (A56) 

 and (A57) that for any integer k ^ 1 



n' ^11 S n" (Ao8) 



From (A54) and (A57) we have 



f F^'-'iw + by/^')fAw) dw = -^ (Ao9) 



where f.iiw) is the density of the normal distribution, truncated at A 

 and —A. The right hand member of (A59) is positive and less than 

 unity since £ < 1 — P*. Hence there exists a Wa with \wa \ ^ A such that 



( F'-'iw + b^^')fAw) dw = F'~\wa + bV^') (A60) 



J— A 



Since Wa is bounded and 7i" is large for large k we can use the well-known 

 approximation 



L V^iriwA + b\/n") J 



_ , (A61) 

 ^ ^^ f_ 0^1) exp [-(wa-{- bVn"f/2\ 



'2^iwA + bVn") i 



where only the leading term is considered. Hence from (A59), (AGO) 



