i.i.d. i.i.d. Let X1, ... ,X*. betalu, 1) and Y1,..., Ym 10. beta(0,1). ) The X's and Ys are also independent. (i) Find

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I I D I I D Let X1 X Betalu 1 And Y1 Ym 10 Beta 0 1 The X S And Ys Are Also Independent I Find 1 (163.66 KiB) Viewed 84 times

i.i.d. i.i.d. Let X1, ... ,X*. betalu, 1) and Y1,..., Ym 10. beta(0,1). ) The X's and Ys are also independent. (i) Find a GLRT for Ho := 0 vs. Hy :u #0. (ii) Show that the test can be based on the statistic T = 2, log X, log Xi+s, log Y, (iii) Find the distributions of – log X; and - L1_, log Xi, and then the null distribution of T under Ho. Give the rejection region of a level a test based on T. Hint: The following facts may be useful: • 1(1) = T(2) = 1, 1(2+1) = x1(x) • beta(a, B) has pdf fbeta (x|0,8) = F&F 24-1(1–2)8–1,0<r<1, ()() 0,3 > 0. In particular, fbeta (2|p, 1) = uak-1. = a • Exponential(e) has pdf fexp(x|u) = 2e-/4 and mean f. • Gammala, j) has pdf fgam (x]a, j) = ner(@24-10-2/4 and mean αμ. • Exponential() is a special case of Gammala, d) with a = 1. • If Xı ~ Gamma(@1,4),..., Xn ~ Gammalan,k), and the X 's are independent, then XT=1 X~ Gamma(21=1 Qi, f). XI- • If X Gamma(a,x), then cX Gamma(a, cu). . If X ~ Gamma(a,1) is independent of Y ~ Gamma(,1), then x*y ~ beta(a,b). N