REDUCING TIME IN RELIABILITY STUDIES 191 



row (say row j) in the table, the observed row vector (r^ — Vi , 

 Ts — Ti , ■ ■ ■ , Vk — z'l) is such that each comyonent is at least as large as 

 the corresponding component of row j. 



Properties of Rs for k = 2 and g = 



For A- = 2 and ^ = the procedure Rs is an example of a Sequential 

 Probability Ratio test as defined by A. Wald in his book.^ The Average 

 Sample Number (ASN) function and the Operating Characteristics (OC) 

 function for Rs can be obtained from the general formulae given by 

 Wald. Both of these functions depend on di and 0-2 only through their 

 ratio a. In our problem there is no excess over the boundary and hence 

 Wald's approximation formulas are exact. When our problem is put into 

 the Wald framework, the symmetry of our problem implies equal proba- 

 bilities of type 1 and type 2 errors. The OC function takes on comple- 

 mentary values for any point a = 61/62 and its reciprocal 62/61 . We shall 

 therefore compute it only for a ^ 1 and denote it by P{a). For a > 1 

 the quantity P(a) denotes the probability of a correct selection for the 

 true ratio a. 



The equation determining Wald's h function is 



1 + a 1 + a 

 for which the non-zero solution in h is easily computed to be 



h{a) = }^ (9) 



In 



a 



Hence we obtain from Wald's formula (3:43) in Reference 5 



s 



a 



Pia) = -^^ (10) 



where s is the smallest integer greater than or equal to 



S = In [PV(1 - P*)]/ln a* (11) 



In particular, for a = 1"^, a* and 00 we have 



Pi^^) = 1/2, ^(«*) ^ P*, P(^) = 1 (12) 



^\'e have written P(l"^) above for lim P{x) as x -^ 1 from the right. The 

 procedure becomes more efficient if we choose P and a* so that *S' is an 

 integer. Then s ^ S and P(a*) = P*. 



Letting F denote the total number of observed failures required to 



