Question: Probability as statistic: Let X1, X2, . .., Xn be an i.i.d. sample from the normal distribution N(M, 02), where u and of are both

Probability as statistic:

Let X1, X2, . .., Xn be an i.i.d. sample from the normal distribution N(M, 02), where u and of are both unknown. Here we have two parameters / and o', so our pa- rameter space is Q = R x [0, co). We shall find the best critical region for testing the null hypothesis Ho : 02 =0% against the alternative hypothesis H1 : 02 # 0%. We set 20 = {(14,02) E12102 =091, 21 = 1(14,03) 60210 7091=12120. (i) Briefly argue that both of the hypotheses are composite. (ii) Compute the maximum likelihood of 20 and show that L(20) = (27 . 06)" 1 2 .02 [(xi-1)2/ 1=1 (iii) Compute the maximum likelihood of 2 and show that -n/2 L(0) = 2n . n-' _(xi -x)2 exp i= 1 (iv) Deduce the likelihood ratio L(20) /L(0) is given by 1 = L(20) 'w \ /2 L(2) "-expl-2+ 21. where w = En(xi -x)2. (v) Briefly discuss why solving L(20) /L(0) nlog(w) - w+ns2logk w

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