1. Suppose that X, X, X, are iid Berwulli(p),0 <p<1. Let U. - x Show that,...
2. Suppose that X1, X2, .. , Xn are iid N(0, 02). Where i and o both assumed to be unknown. Let 0 = (i,a). Find jointly sufficient statistics for 0
Let X1, X2,· · ·iid B(1, x), i.e,P(X1= 1) =x= 1−P(X1= 0), where x∈ [0,1]. Let Sn = X1+X2+· · ·+Xn. What can you say about the limiting behaviour of Sn/n from strong law large number
7.5.12 Suppose that X.., X, are iid Bernoulli(p) where 0<p s an unknown parameter. Consider the parametric function T(p)-p + qe with q p. (i) Find a suitable unbiased estimator T for (p); (ii) Since the complete sufficient statistic is = Ση!Xi, use the Lehmann-Scheffé theorems and evaluate the conditional expec tation, E [I, I u-11]; (iii) Hence, derive the UMVUE for T(p) Hint: Try and use the mgf of the Xs appropriately.) 7.5.12 Suppose that X.., X, are iid...
Let X1, · · · ,Xn be iid from Uniform(−θ,θ), where θ > 0. Let X(1) < X(2) < ... < X(n) denotes the order statistics. (a) Find a minimal sufficient statistics for θ (d) Find the UMVUE for θ. (e) Find the UMVUE for τ(θ) = P(X1 > k).
2. Let X1, X2, X3 ..., X, be iid b(1, p) random variables. Let Sn = 27-1Xthen prove that Sn-E(Sn) N(0,1) as n +00. (Sn)
4.(120) Let X1,,,Xn be iid r(, 1) and g(u) given. Let 6n be the MLE of g(4) (1)(60) Find the asymptotic distribution of 6, (2)(60) Find the ARE of T Icc(X) w.r.t. on P(X1> c), c > 0 is i n i1 5.(80) Let X1, ,,Xn be iid with E(X1) = u and Var(X1) limiting distribution of nlog (1 +). o2. Find the where T n(X - 4)/s. - 1 - 4.(120) Let X1,,,Xn be iid r(, 1) and g(u)...
, xn is an iid sample from fx(x10)-θe-8z1(x > 0), where θ > 0. Suppose X1, X2, For n 2 2, n- is the uniformly minimum variance unbiased estimator (UMVUE) of 0 (d) For this part only, suppose that n-1. If T(Xi) is an unbiased estimator of e, show that Pe(T(X) 0)>0
15. Suppose Ui ~ iid Unif(0, 1) for n = 6. Let X = U(1), Y = U(6), and W = X/Y. Find: ~Ll b) Fw(w) c) E(W) d) Var(W)
7. Let X1,... , Xn be iid based on f(x; 6) -22e-z?/e where x > 0. Show that θ=-yx? is efficient
- Let X1, X2, ..., Xn be iid from the pdf fe(x) = 0e-82, > 0. Note that T = 2 , X, is a sufficient statistic. Consider testing the hypothesis H.: 8 = 1 vs H: 8 = 2 using Bayes method. Suppose the prior distribution is P(0 = 1) = ? and P(0 = 2) = 1 - . (a) Show that the Bayes test rejects H, if T < In log(2) + log((1 - ))) (b) Take...