SELECTING THE BEST ONE OF SEVERAL BINOMIAL POPULATIONS 571 



in Figs. 2 through 5. As d* —* both the normal approximation and the 

 true value are asymptotically equivalent to the straight line approxima- 

 tion. 



The normal approximation to the probability of a correct selection 

 can also be written in another form similar to (A16) which is actually 

 more useful for numerical calculations. The left member of (A21) can 

 be written as 



< ^^^1 Vp[i]g[ii + (P[i] - P[i]) Vn X] p ,j^^ ^ ^^^ 



Vp[i]q[i] iJ 



where 



TTr -X"(t) — np[i] / KnA\ 



Wi = / it = 1, 2, • • • , k) (A34) 



and Wi is the same function of X(i) as Wi is of X(i) . The outside summa- 

 tion in (A33) is over the values taken on by wi as x^d runs from to n. 

 As n ^ <x the expression in (A33) approaches 



Pcs= ' ' ' I '^ ''"^ Vp[i]q[i] + (?>[ii - P[i]) Vn 



i:[pc- 



Vpami] 



f{w)dw (A35) 



where J{t) is the standard normal density and F{t) is the standard normal 

 c.d.f. For the symmetric configuration, which is least favorable for large 

 n, (A3o) reduces to 



Pes ^ r F'-' (w + -f ^^ \ fiw) dw (A36) 



^-^ \ Vi - d*y 



A straightforward integration by parts gives the alternative form 



2d* V~n ^ 



''—00 L 



1 - F[iv - 



Vi - rf*v- 



F' -{w)f(w)dw (A37) 



which corresponds to (A20). 



A simple method for computing such integrals based on Hermite poly- 

 nomials is described by Salzer, Zucker, and Capuano. 



LARGE SAMPLE THEORY — ALTERNATIVE SPECIFICATION 



The expression corresponding to (A22) for the alternative specification 



IS 



pL, > - /Pm-Pr2])v g (^• = 1, 2, . . . , k - 1)) (A38) 



^ vpfi]?*!] + Pr.]q[2] i 



