Math 234 Section 1Homework 6 - Spring 2020AsymptoticsDue: Mar 26, 2020 11:59 pm.Instructions: Please write neat solutions for the problems below. Show all your work. If necessary,explain your solution in words. If you only write the answer with no work, you may not be givenany credit.Please submit your entire homework as a single pdf file. Use pdf merging tools as necessary. Forproblems 1-3, you can scan your responses using a scanner or a phone scanning app. You are notallowed to simply take a photo of your homework due to poor lighting. The grader reserves theright not to grade your submission if it is unclear.Problems1. Suppose that X1, . . . , Xn are iid with common density functionf(x | θ) = 12(1 + θx), −1 ≤ x ≤ 1for −1 ≤ θ ≤ 1. Find a consistent estimator of θ. Justify that the estimator is consistent. Hint: Find theMOM estimator.2. Let X1, . . . , Xn be iid with mean µ and variance σ2 whose second derivative g00 is continuous with g00(µ) 6= 0. Recall that X¯n =1nPni=1 Xi.(a) Show that √n|g(X¯n)−g(µ)| → 0 in distribution as n → ∞. Note that this is different from √n(g(X¯n)−g(µ)).(b) Show that √n|g(X¯n) − g(µ)| → 0 in probability as n → ∞.(c) Show that n[g(X¯n) − g(µ)] converges in distribution to 12g00(µ)σ2χ1 as n → ∞.(d) Use the previous result to show that when µ = 0.5,n[X¯n(1 − X¯n) − µ(1 − µ)] → −σ2χ21in distribution as n → ∞.Hint: For part a), mimic the proof of the Delta method showed in class, i.e. start by using the MeanValue Theorem. The Central Limit Theorem and Slutsky’s Theorem (Thm 5.5, p. 75 of Wasserman) maybe useful. For part b), look at Thm 5.4c of Wasserman (this was also discussed in lecture). For part c),construct a 2nd-order Taylor series expansion of g at µ = X¯n. You can ignore the higher-or代写Math 234作业、代做data留学生作业、代做R语言作业、代写R程序设计作业 代做Python程序|代做留学生Pdered termsin the Taylor series expansion for this problem. CLT and Slutsky’s theorem may be handy in completingyour proof.3. Continuation of Problem 3 of Homework 5. Suppose that X1, . . . , Xn are iid with distribution U(0, θ),θ > 0 is an unknown parameter. In class, we showed that ˆθn = max(X1, . . . , Xn) is the MLE. Perform thefollowing.(a) Show that ˆθn is consistent. You may freely quote the results derived in the previous homework.(b) Show that the limiting distribution of −n(ˆθn − θ) as n → ∞ is exponential. Hint: you may need torefresh on L’hopital’s rule for evaluating limits.(c) Give an approximate value for Var(ˆθn) for large n.(d) For this example, we obtain that the MLE is asymptotically exponential instead of asymtptoticallynormal as we demonstrated in lecture. Briefly explain why this does not contradict what we establishedin class. Hint: Look at the sketch of the proof for asymptotic normality of MLE. We took the derivativeof the log likelihood function with respect to θ. Can you do that here?4. R exercise. The objective of this exercise is to learn how to use R to perform bootstrap.(a) Read p. 187-190 of the Intro to Statistical Learning book. This is identical to the lecture on bootstrappingbut with more details.(b) Read p. 194-195 of the book. Only read the section “Estimating the Accuracy of a Statistic ofInterest”. Skip the part on the linear regression model.(c) Work on (a), (b), (c) of Problem 9, p. 201. For part (b), an estimate of the standard error is givenby √snwhere s is the (observed) sample standard deviation. For part (c), generate at least 1000bootstrap samples. In addition, plot a histogram of the bootstrap samples you generated. This linkon plotting histograms may be helpful. Include your responses to a pdf file that you must submit toNYU classes.转自：http://www.6daixie.com/contents/18/5013.html