Question: S Test against decreasing trend. Consider the infinite product model for the 2-test of goodness of fit, and let be the uniform distribution on E

S Test against decreasing trend. Consider the infinite product model for the 2-test of goodness of fit, and let be the uniform distribution on E D ¹1; : : : ; sº. Suppose the null hypothesis H0 W # D is not to be tested against the full H1 W # 6D , but only against H0 1

W #1 > #2 > > #s (‘decreasing trend’). The 2-test is then not particularly useful

(why?). It is better to use the test statistic

T = hn(i)-n(s+1)/2 n(s2-1)/12

Calculate E#.Tn/ and V#.Tn/ and show that Tn !d N0;1. Use this result to develop a test of H0 against H0 1. Hint: Represent Tn as a sum of independent random variables.

T = hn(i)-n(s+1)/2 n(s2-1)/12

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