The Sj-test against linear trend

The Sj-test proposed by Noether [7] is a sequentialtest of the hypothesis of randomness against the alter-native of linear trend, which can be expressed as thehypothesis that the joint distribution of xl, x2, ••• , lhis given byifi=lThe Sj-test is carried out by comparing Xi with Xi+j' forfixed j ~ o, and using the sequential probability ratiotest of the hypothesis H0: p = P(Xi~ Xi+j} =~againstthe alternative H1 : p = ~ + fj, the test being truncatedwith j comparisons if a decision has not been reached bythat point.To carry out the Sj-test the value of j is chosen tominimize the expected numbers of observations required bythe sequential test. The dissertation provides a chartfrom which one can read the value of j required for givenalternatives 9 for o( = 13 = 0.05 and ot = t3 = 0.01,where 0( , S represent the probabilities of type I,type II error respectively. Since the Sj-test is carriedout as a truncated sequential test, the effect of trunca-tion on the probabilities of error and on the averagesample number is considered.83