40. Suppose that X,.. , X N(0,o?). (a) Determine the asymptotic distribution of the iid re...
Let X1 , . . . , xn be n iid. random variables with distribution N (θ, θ) for some unknown θ > 0. In the last homework, you have computed the maximum likelihood estimator θ for θ in terms of the sample averages of the linear and quadratic means, i.e. Xn and X,and applied the CLT and delta method to find its asymptotic variance. In this problem, you will compute the asymptotic variance of θ via the Fisher Information....
Problem 5: Variance stabilizing transformations are transformations for which the re- sulting statistic has an asymptotic variance that is independent of the parameters of interest. Let T = 3 Xị, where X, Poisson(1). (a) Find the asymptotic distribution of In, that is, the limiting distribution of Vn(vTn – V), and show that it is variance stabilizing (If Vn(vTn - 4 N(0,02), then o2 is called the asymptotic variance of VTn). (b) Give an approximate mean and variance of Tn for...
valu Exercises 8.2. x, . . . ,x, nd G(p), the geometric distribution with mean 1/p. Assume that e size n is sufficiently large to warrant to invocation of the Central Limit Theo- . Suppose se that Xi , . . . X, Use the asymptotic d confidence interval for p Suppose that XN(0, o2) (a) Obtain the asymptotic distribution of the second istribution of p 1/X to obtain an approximate 100(1-u)% Suppose sample moment m2 -(I/n)i X. (b) Identify...
Part 1: Derive the expected value and find the asymptotic
distribution.
Part 2: Find the consistent estimator and use the central limit
theorem
b. Derive the expected value of X for the Weibull(X,2) distribution. c. Suppose X,.. .X,~iid Uniffo,0). Find the asymptotic distribution of Z-n(-Xm) max Let X, X, ~İ.id. Exp(β). a. Find a consistent estimator for the second moment E(X"). Use the mgf of X to prove that your estimator is consistent in the case β=2 b. Use the...
5.4.3 Suppose that X,, X" are iid exponential with mean β(> 0), , y, are iid exponential with mean η(> 0), and that the A's are independent of the Ys. Define ½,-. Then. m, n (i) Show that I , is distributed as /m (ii) Determine the asymptotic distribution of T m, n 2m, 2n as n ->o, when is m, n kept fixed; (ii) Determine the asymptotic distribution of T, as m, when n is kept fixed.
Let Xi iid∼ N(0, θ) for i = 1, ..., n.
a) Find the MLE for θ. Call it
b) Is biased?
c) Is
consistent?
d) Find the variance of
(e) What is the asymptotic distribution of ?
22. Suppose that X1, X2,...,x, Fp, where F, is a discrete distribution with probability mass function p(x) = p(1– p)* for x=0,1,2,.... (See Example 7.3.9.) The MLE Ộ=1/(1+X) has the asymptotic distribution below: VnCô–p) DY~N(0,(1 – p)p?). (8.54) Use the normal distribution in (8.54) to obtain, via a variance stabilizing transformation, an approximate 100(1 – a)% confidence interval for p.
12. Suppose XIX, iid X, P(θ, l), where P(0,1) is the one-parameter Pareto distribution with density f(x)-0/10+1 for l < x < 00, Assume that θ >2, so that the model θ/(0-1)(8-2)2 (a) obtain the MME θι from the first moment equation and the MIE θ2 (b) Obtain the asymptotic distributions of these two estimators. (c) Show that the ML is asymptotically superior to the MME P(0,1) has finite mean θ/(9 -1 ) and variance
2. Suppose X ~ N (μ,5). Find the asymptotic distribution of X(1-X) using A-methods. 3. Let X denote that the sample mean of a random sample of Xi,Xn from a distribution that has pdf Let Y,-VFi(x-1). Note that X = lari Xi- (a) Show that Mx(t) = (ca-tryM f(x) = e-z, x > 0. Find lim+oo My, (t)
Problem 4 Define f(x) as follows θ2 -1<=x<0 1-θ2 0<=x>1 0 otherwise Let X1, … Xn be iid random variables with density f for some unknown θ (0,1), Let a be the number of Xi which are negatives and b be the number of Xi which are positive. Total number of samples n = a+b. Find he Maximum likelihood estimator of θ? Is it asymptotically normal in this sample? Find the asymptotic variance Consider the following hypotheses: H0: X is...