144 Agricultural Research and Productivity 



y^ Yield, technological level, net income in time t 



(/ = 0, 1,2,...), the 5/a/6' variable 



n^ Number of trials in period ^ « = 1, 2, . . . the 



control variable 



c{n) Cost of experimentation, withc(O) =0, c\n) 



> 0, c"(n) < 



X. Yield in trial /,/= 1, 2, .. .,A7 



fix) Probability density function of x 



F(x) Cumulative distribution of X 



z The largest value in a sample of the random 



I variate x 



Oi = . I The discount factor with r the rate of interest 



V The objection function 



E[ ] Expectation operator 



^y — yt — yt-i Yield increment 



^[A;^] Mean of A>^ 



Var (A>^) Variance of A v. 



Search 



The search process is a sequence o{ experiments, each composed of 

 n, trials. A single trial can be a test of a technique — one variety of 

 a crop, a certain dose of fertilizers, one planting date, etc. Because 

 of the variability in experimental conditions, a trial is usually car- 

 ried out in a number of replications. This variability is disregarded 

 here, and it is assumed that a trial has a single outcome. 



Two slightly different ''stories'' form the background for two 

 versions of the statistical description of the search procedure. 

 Both versions involve drawing a sample from a random distribu- 

 tion and selecting a value. In one of these versions the value 

 chosen will be the largest in the sample, in the other it will be the 

 average. The statistical process of choosing the largest outcome 

 from a set of random drawings is treated under the heading of the 

 theory of extreme values (Epstein 1960, Gumbel 1958) in the 

 general subject of order statistics. 



Utilizing the symbols introduced earlier, x, is the yield in trial /, 

 and z = Xj, Xj > Xj (i = 1,2,... n). 



The cumulative distribution of z is 



HJz) = Fr (-dW X. < z) - F"(z) (8.1) 



