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# 这是一篇来自美国的关于通过代码解决下面一些问题的代写

Group discussion of homework problems is allowed, but the fifinal write-up should be individual. For each problem please show calculations and/or provide an explanation to justify your answer. For the computing parts, use the provided RMarkdown template fifile to submit your code and fifigures.

Problem 1 (10 points)

Exercise 7.2. For part b, you may use an offff-the-shelf optimization solver, e.g.

R’s optim function.

Problem 2 (5 points)

Exercise 7.4

Problem 3 (10 points)

Exercise 7.6 (b), (c)

Problem 4 (15 points)

Simulate N = 1000 independent random samples of sizes n = 10, 100, 1000, 10000 from the pdf given in Exercise 7.6 with θ = 1.

(a) Plot a separate histogram of the MLEs computed over the 1000 random samples for each of the values of n.

(b) Repeat part (a) for the MOM estimator.

(c) Plot the average squared error for the MLE as a function of n. The average squared error is given by

1/N N Xk=1(θ ˆ k(X1, . . . , Xn) θ) 2 ,

where θ ˆ k(X1, . . . , Xn) is the MLE of the kth random sample of size n.

(d) Repeat part (c) for the MOM estimator.

(e) Brieflfly discuss the similarities/difffferences between the MLE and MOM based on your simulation results. Put your discussion in the RMark

down fifile.

Problem 6 (10 points)

Exercise 7.9

Problem 7 (15 points)

Let X be a continuous random variable with cdf FX and pdf fX such that E|X| < .

• Prove that E|X a| < for all a.
• Prove that for all a

E|X a| ≥ E|X m|

where m is the median of X, i.e., FX(m) =12. In other words, the median minimizes the function g(a) = E|X a|.

Problem 8 (15 points)

Let X be a continuous random variable with cdf FX and pdf fX such that E|X| < . Let τ (0, 1) denote a quantile and defifine the “check-loss” function as

ρτ (u) = ((τ 1)u i fu < 0

τu ifu 0.

• Prove that E[ρτ (X a)] < for all a.
• Prove that for all a

E[ρτ (X a)] E[ρτ (X qτ )]

where qτ is the τ th quantile of X, i.e., FX(qτ ) = τ . In other words, the τ th quantile minimizes the function g(a) = E[ρτ (X a)]. 