5. Let Xn ~ χ2 (n), n-1, 2, . . . . Find the limiting distribution...
8(100) Let X1,,Xn be iid from r(a, 6). (1)(50) Find the limiting distribution of the MLE of B. (2)(30) Find the limiting distribution of the MLE of B when a is known. (3)(20) Compare two asymptotic variances in (1) and (2), and make comment on it. 1ラ
8(100) Let X1,,Xn be iid from r(a, 6). (1)(50) Find the limiting distribution of the MLE of B. (2)(30) Find the limiting distribution of the MLE of B when a is known. (3)(20)...
Let Yn be a chi square random variable with n degrees of freedom, and let Xn = Yn / n2. Find the limiting distribution of Xn.
Let X1, X2, .. , Xn be a random sample of size n from a geometric distribution with pmf =0.75 . 0.25z-1, f(x) X-1.2.3. ) Let Zn 3 n n-2ућ. Find Mz, (t), the mgf of Žn. Then find the limiting mgf limn→oo MZm (t). What is the limiting distribution of Z,'?
Let X1, X2, .. , Xn be a random sample of size n from a geometric distribution with pmf =0.75 . 0.25z-1, f(x) X-1.2.3.
) Let Zn 3...
Let X1, X2, ..., Xn be a random sample with probability density
function
a) Is ˜θ unbiased for θ? Explain.
b) Is ˜θ consistent for θ? Explain.
c) Find the limiting distribution of √ n( ˜θ − θ).
need only C,D, and E
Let X1, X2, Xn be random sample with probability density function 4. a f(x:0) 0 for 0 〈 x a) Find the expected value of X b) Find the method of moments estimator θ e) Is θ...
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)...
6. Let X1,..., Xn be a random sample from Uniform (0, 1). a) Find the exact distribution of U = – log(X(1)) where X(1) = min(X1, X2,..., Xn). b) Find the limiting distribution of n(1 – X(n)), where X(n) = max(X1, X2, ..., Xn).
(x-2) 5. a) Let S Prove that s? Po? n-1 b) Consider a sequence of random variables {Xn} with pdf, fx, (x) = xht where 1<x<. Obtain Fx (2) and hence find the limiting distribution of X, as noo. c) Consider a random sample of size n from Fx (x) = where - <I<0. Find the limiting distribution of Yn as n + if (a)' = n max{X1, X2, X3,...,xn). and X(n) [17 marks]
Let Zn BE X2(n) (Chi-square) and let Wn = Zn/n2. Find the limiting distribution of Wn using the Weak Law of Large Numbers
Let X1, . . . , Xn be a sample taken from the Gamma distribution Γ(2, θ−1) with pdf f(x,θ)= θ^2xexp(−θx) if x ≥ 0, θ ∈ (0,∞), and 0 otherwise, (A) Show that Y = ∑ni=1 Xi is a complete and sufficient statistic. (B) Find E(1/Y) . Hint: If W ∼ χ2(k) then E(W^m) = 2mΓ(k/2+m) for m > −k/2. Note also that Y Γ(k/2) Γ(n) = (n − 1)!, n ∈ N∗ . Facts from 1(C) are useful:...
2) Let X,..X, be ii.d. N(O, 1) random variables. Define U- Find the limiting distribution of Zn (Hint: Recall that if X and Y are independent N(0, 1) random variables, then has a Cauchy distribution
2) Let X,..X, be ii.d. N(O, 1) random variables. Define U- Find the limiting distribution of Zn (Hint: Recall that if X and Y are independent N(0, 1) random variables, then has a Cauchy distribution