Question 8. Consider a sequence of n independent trials, each resulting in one of k + 1 outcomes 1, 2, ……· ‚k + 1. Outcome j occurs with probability p; on any given trial. Let Y; be the number of trials resulting in outcome j. Consider testing the simple null hypothesis 1, 2, …… · ‚ k + 1. Find the likelihood ratio statistic An and show that 0 where Pj = Пј for j -2 log(n) - Qn = P k+1 Qn => (Yj – nπj)² Nπ j Using the asymptotic equivalence of -2 log(n) and Qn find the asymptotic distribution of Qn. (Hint: Use the Taylor expansion f(x) = x log(x/x0) = (x − x0) + (x − xo)²/2xo + o[(x − x0)²]. ) - Solution: -