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Stat 462/862 Assignment 4
(Due in my mailbox at Jeffery Hall 406 on Dec 5h, 2019)
1. This problem involves the OJ data set which is part of the ISLR package.
(a) Create a training set containing a random sample of 800 observations, and a test
set containing the remaining observations.
(b) Fit a tree to the training data, with P urchase as the response and the other variables
as predictors. Use the summary() function to produce summary statistics
about the tree, and describe the results obtained. What is the training error rate?
How many terminal nodes does the tree have?
(c) Create a plot of the tree, and interpret the results.
(d) Predict the response on the test data, and produce a confusion matrix comparing
the test labels to the predicted test labels. What is the test error rate?
(e) Apply the cv.tree() function to the training set in order to determine the optimal
tree size.
(f) Produce a plot with tree size on the x-axis and cross-validated classification error
rate on the y-axis.
(g) Which tree size corresponds to the lowest cross-validated classification error rate?
(h) Produce a pruned tree corresponding to the optimal tree size obtained using crossvalidation.
If cross-validation does not lead to selection of a pruned tree, then
create a pruned tree with five terminal nodes.
(i) Compare the training error rates between the pruned and unpruned trees. Which
is higher?
(j) Compare the test error rates between the pruned and unpruned trees. Which is
higher?
2. Consider the problem of generating sample from a Beta distribution Be(α, β).
(a) One result is, if two Gamma random variables are X1 ∼ Ga(α, 1) and X2 ∼
Ga(β, 1), then X =X1X1 + X2∼ Be(α, β).1
Use this result to construct an algorithm to generate a Beta random sample.
Provide a density histogram to emulate the performance.
(b) Compare the algorithm in (a) with the rejection method based on (i) the uniform
distribution; (ii) the truncated normal distribution.
3. Consider estimating the integral
θ =Z ∞0exp(−(√x + 0.5x))sin2
(x)dx
where the pdf of x is f(x) = 0.5 exp(−0.5x).
(a) Conduct the Monte Carlo (MC) integration for estimating θ.
(b) Conduct MC integration using importance sampling with the following proposal
functions
g1(x) = 1
2
exp(−|x|),(Laplace Distribution)
g2(x) = 12π11 + x2/4,
g3(x) = 1√2πexp(−x2/2).
For sample size M = 100, 500, 1000, 2000, compare the mean and standard deviations
of the estimates.
(c) (For graduate students 862 only) Implement MC integration using self-normalized
importance sampling with g(x) from a mixture normal density. Explain the procedure
and integrate your results clearly.
4. (a) Provide a Metropolis-Hastings algorithm to generate samples from a binomial
distribution Bino(n, p) with
P(X = k) =nk!pk(1 − p)n−k, k = 0, . . . , n.
Use uniform distribution in {0, . . . , n} as proposal distribution and use independent
chains. Compare estimated means and variances with the known theoretical
means and variances of the binomial distribution.
2
(b) Provide a Metropolis-Hastings algorithm to generate samples from a standard normal
distribution. The proposal distribution is the normal distribution with the
mean being the current value in the chain and the variance being 0.25,0.01,100,
respectively. Compare the estimated means and variance with the known theoretical
means and variance of a standard normal distribution.
3

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