Monte Carlo integration

Problem 1 (5 points)
The following R code implements a Monte Carlo integration:
n = 1000
U = runif(n)
X = U/(U^2+1)
print(sum(X)/n)
As pseudocode, this says

1. Set n=1000
2. Generate n samples from U(0,1)
3. For each U(i), we let X(i) = U(i)/[U(i)^2 + 1]
4. Add up all values of X, divide by n
5. Print the result
   In either case, the output is: 0.3388646
   What is the asymptotic value of this test? (The limiting value as n→ ∞?)
   Problem 2 (15 points)
   On any given day, the probability that the power goes out in a certain neighborhood is 0.004
   (or 1/250). Assuming that each day is independent, find the probability that there are no
   power outages for 1000 consecutive days using
   (a) The binomial distribution.
   (b) An approximation using the Poisson distribution.
   (c) An approximation using the normal distribution.
   Problem 3 (5 points)
   In Exercise 5.1.7, we considered a collection of random variables with the shifted exponential
   distribution. That is, we let X1, . . . , Xn be iid random variables with common pdf
   f(x) = �
   e−(x−θ) x > θ, −∞ < θ < ∞ 0 elsewhere. We showed that when Yn = min{X1, . . . , Xn}, then the cdf of Yn is FYn (t) = � 0 t < θ 1 − �eθ et �n t ≥ θ Given this cdf, show that Yn − 1 n is an unbiased estimator of θ. Hint (1): Integration by parts shows that for a constant k �= 0 � xekxdx = 1 k xekx − 1 k2 ekx + C Hint (2): L’Hˆopital’s rule shows limx→∞ xe−x = limx→∞ x ex = limx→∞ 1 ex = 0 Problem 4 (10 points) Let X be the mean of a random sample of size n from a N(θ, σ2) distribution, −∞ < θ < ∞, σ2 > 0. Assume that σ2 is known.
   (a) Find the Fisher information I(θ).
   (b) Show that X is an efficient estimator of θ.
   Problem 5 (5 points)
   Let X1, X2, . . . Xn be a random sample from a Poisson distribution with mean θ > 0. The
   mle of θ is ˆθ = x and Fisher’s information is I(θ) = 1
   θ . Construct a Rao’s score statistic test
   of the hypothesis H0 : θ = 3 versus H1 : θ �= 3.
   Note: You do not need to derive the mle and Fisher’s information for the Poisson distribution.
   Problem 6 (10 points)
   Consider a sample where X ∼ N(θ, 1). If n1 samples are observed, n2 samples are censored
   at level a, then for the EM algorithm, if ˆθ(m) is the EM estimate on the mth step, the (m+1)
   step estimate is
   ˆθ(m+1) = n1
   n
   x +
   n2
   n ˆθ(m) +
   n2
   n
   φ(a − ˆθ(m)
   )
   1 − Φ(a − ˆθ(m))
   (This is Equation 6.6.16 in our text.)
   a) Suppose that n2 = 0; that is, there are no censored values. Suppose we choose an initial
   guess of ˆθ(1) = k. What is the value of ˆθ(2)? What is the value of ˆθ(3)?
   b) Suppose instead that n1 = n2, x = 0, a = 0, and ˆθ(m) = 0. What is the value of ˆθ(m+1)?