Two sequential tests against cyclic trend
Roberts, Helen Murray
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Let the chance variables x1, x2, •••, xn have the joint cumulative distribution F: F(x1, x2,•••,Xn) and assume that the distribution function F(x1, x2,•••,xn) is continuous. Let ^n be the class of all continuous cumulative distribution functions. Let Wn be the class of all continuous cumulative distribution functions of the form F(x1,x2,•••,Xn) = F(x1)F(x2)•••F(Xn). The hypothesis of randomness states that F(x1,x2,•••,xn) assumed to belong to ^n actually belongs to Wn. In this dissertation two sequential tests of randomness proposed by Noether are studied. In the first sequential test the alternative to randomness is characterized by a stochastic relation of the type Xi= Xi-l + Ui, in the second sequential test the alternative is characterized by an irregular cyclical trend. The first test is based on the statistic Tm which is equal to the number of rank positions xm+1 may take given the ranks of (x1,x2,•••,xm) so as to convert (z1,z2,•••,zm-1) into (z1,z2,•••,zm) where zi=sign(xi+1 - xi). It is shown under the null hypothesis that Tm is an unbiased estimate of a corresponding population parameter Tm and is a biased estimate of Tm under the alternative hypothesis. The properties of Tm under the null hypothesis are then examined and it is shown that Tm and Tm+k (k > 2) are independent. It follows from this property, by using Hoeffding and Robbins theorem, that sigma log Ti is asymptotically normal. It is shown, both under the null and the alternative hypotheses, that this test terminates with probability one. By sampling some idea is gained about the number of observations needed for the test to terminate under both hypotheses, and also about the effect of this modified test on the probabilities of Type I and Type II errors. The second sequential test is based on runs up-and-down. This test is described and then a modification of this test is studied. Runs of three different lengths are considered and the corresponding parameter determined. By sampling some idea is obtained about the number of observations needed for the modified test to terminate, and the effect of this test on the probabilities of Type I and Type II errors. The first seauential test, the sequential run test and the modified sequential run test are compared.