Question 7. Generalized Neyman-Pearson Lemma. Let fo(x), f1(x), · ‚ ƒk(x) be k + 1 probability density functions. Let 0 be a test function of the form 1, k j=1 0(x) = (x), 0, if fo(x)>ajf; (x) if fo(x) =Σajf; (x) if_fo(x) < Σ½³±1 αjƒ¡(x) where a; 0 for j = 1, .. ,k. Show that 0 maximizes among all , 0 ≤ ≤ 1, such that Solution: 2)f(x)dz (x)dx [ ø(x)f;(x)dr ≤ [ Þ(x)ƒ,(x)dx, j = 1, 2,..., k

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