SELECTING THE BEST OXE OF SEVERAL BIXOML\L POPULATIONS 541 



tions. Then the confidence statement, consisting of two sets of inequah- 

 ties 



P[i] - d* ^ ps ^ p 



\ 



11] 



^PV2] S Pu ^ Pvi\ + d*^ 



or 



^ p[i] — ps ^ d* 

 S Pu - 7^12, ^ ci*^ 



has confidence level P*. It should be noted that the above confidence 

 statement is not a statement about the value of any p but is a statement 

 about the correctness of the selection made. 



LEAST FAVORABLE CONFIGURATION 



The main idea used in the construction of the tables was that of a 

 least favorable configuration. Before defining this concept we shall 

 define the set of configurations 



Pm - d = p[2\ = Pm = • • • = pik] (6) 



obtained by letting d in (6) varj^ o\'er the closed interval {d*, 1) as the 

 Less- Favorable set of configurations. It is intuitively clear and will be 

 rigorously shown in Appendix II that if our procedure satisfies the 

 specification for any true configuration (6) with d = d and p[i] = p^, 

 then it will also satisfy the specification when 



pli] - / ^ Pm ^ Pi?] ^ • • • ^ P[k] (7) 



Of course, we shall be interested particularly in the case in which d 

 equals the specified value d*. If d = d* is fixed in (6), then (6) specifies 

 the differences between the p- values, but the '"location" of the set is still 

 not specified. We shall use p[i] to locate the set of p-values. The proba- 

 bility Pes of a correct selection for configurations like (6) with d — d* 

 depends not only on d*, n and k but also on the location pm of the 

 largest p- value (except for the special case h = 2 and n = 1). [In the 

 corresponding problem for selecting the largest population mean of k 

 independent Normal distributions with unit variance," this probability 

 Pes depends only on the differences and, hence, only on (/ in the configura- 

 tion correspondhig to (6)]. 



When (6) holds with any fixed value of c?, the probability Pes (for any 

 fixed n) may be regarded as a function of p[ii where d ^ p[\] ^ 1). This 

 function is continuous and bounded over a closed interval and therefore 

 assumes its minimum value at some point p[i] {d) = p[i\ {d;n) in the 

 closed interval {d,\). Fig. 1(b) gives the value of p\i] (d) as a function 

 of d for A- = 3 and for 7i = 1, 2, 4, 10 and x . For any particular value 



