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日期:2020-02-12 08:37


STAT 440

Homework #3

Goal: Importance sampling, bootstrap, hypothesis testing

For this homework, submit your R code for this assignment electronically

via Canvas. Note the deadline on Canvas. In addition to the R code, you

must also turn in your typed-in answers. Please submit your R code

and the typed up answers as separate documents. Points will be

deducted if R code is very poorly formatted.

1. Monte Carlo basic

The Weibull distribution is often used to model extreme values. A

Weibull random variable with shape parameter k and scale parameter

λ has density given by

f(x) = kλ xλ(k 1)

exp · xλk , for x > 0 (1)

(a) Find the expected value for the random variable X if X ~ W eibull(k = 3, λ = 5) by using Monte Carlo. The R command for generat?ing from a Weibull distribution is rweibull. Use a Monte Carlo

sample size of 1000.

(b) Find the Monte Carlo standard error for your Monte Carlo es?timate from above. Report a 95% confidence interval for your

estimate.

(c) Now find the Monte Carlo estimate of the expected value of X

using a sample size of 100, 000. Again, report your estimate along

with its Monte Carlo standard error and a 95% confidence inter?val. How do the new estimate and confidence interval compare

to those for a sample size of 1000?

(d) Find P(X > 5) using Monte Carlo. Again, report the Monte

Carlo standard error and a 95% confidence interval.

2. Importance sampling

The Pareto pdf is

f(x) = βαβ xβ+1 , a < x < ∞, α > 0, β > 0 (2)

1

Suppose X ~ Pareto(α = 3, β = 5). Use importance sampling to

estimate E(X) and E(X2

). Use as the importance function (proposal

distribution), Gamma(15, 0.25). Generate 10,000 samples from the

Gamma(15, 0.25) pdf to obtain your estimate. Use the Gamma pdf

parameterization, f(x) 1

Γ(α)βα xα 1e

x/β. You may use the R command

rgamma to generate draws from the Gamma pdf. Be careful to make

sure the parameterization of the Gamma pdf in R is the same as the

one you are using: in R the second parameter is by default 1/β.

(a) Clearly describe your algorithm in “pseudo-code”, i.e. write out

your algorithm for this problem briefly, systematically, in words,

filling in mathematical details where necessary.

(b) Report your estimates along with the Monte Carlo standard er?rors of your estimates.

3. Now re-use the samples you just obtained (from the Gamma pdf)

above to estimate E(Y ) and E(Y 2

) for Y ~ Pareto(α = 5, β = 7).

Report your Monte Carlo standard errors. Notice how you can change

the distribution you are interested in without generating any new sam?ples.

4. Re-estimate E(X) and E(X2

) for X ~ Pareto(α = 3, β = 5) using a

different importance function. You are welcome to choose any impor?tance function you like, but you have to select one that works better

than the importance function I provided above. Think about what it

means to “work better.”

(a) Report your estimates along with Monte Carlo standard errors.

(b) Justify why you think the importance function you are using here

is better than the one I provided (the Gamma pdf).

5. Inference for Poisson expectation (λ) using maximum likelihood. Sup?pose you have 50 independent realizations of (observations from) a

Poisson(λ) distribution.

(a) Find the MLE of λ using the data provided on Canvas:

hw3 prob5 dat.txt.

(b) Find the 95% confidence interval for λ using standard asymptotic

theory (Central Limit Theorem).

2

(c) Now suppose you want to do a hypothesis test for the null hypoth?esis that λ = 3 versus the alternative the λ < 3. Use standard

asymptotic theory to conduct the hypothesis test and find the

p-value. What is your conclusion?

(d) Now find the exact p-value for the above test using Monte Carlo

(you will no longer use an asymptotic approximation). Report

your Monte Carlo estimate of the p-value and report the Monte

Carlo standard error for this estimate.

6. Bootstrapping

Estimate the mean μ of a population based on a random sample from

that population. Download the samples “hw3 prob6 dat.txt” on Can?vas. The following are different ways to estimate the sampling distri?bution of your estimate.

(a) Write down the estimate, the sample mean Xˉn. Calculate an

estimate of its standard error. Use standard asymptotic theory

(the Central Limit Theorem) and report a 95% confidence interval

for μ based on the standard error estimate.

(b) Now find the approximate sampling distribution of Xˉn using a

non-parametric bootstrap with B = 1000 bootstrap replications.

You should display a clearly labeled histogram of the sampling

distribution of Xˉn.

(c) Using the bootstrap replication from the previous part, estimate

the standard error of your estimate.

(d) You can now calculate approximate 95% confidence intervals for

μ in two ways: (i) use the bootstrap estimate of standard er?ror and use usual asymptotic theory to calculate a 95% confi-

dence interval for μ based on this estimate, and (ii) use a the

25.th and 95.5th percentiles from your bootstrap samples. You

will need to use the command quantiles, for e.g. if your boot?strap samples of Xˉn are in the vector samplemeanboot, type

quantiles(samplemeanboot, c(0.025, 0.0975)).

(e) Now find the approximate sampling distribution of Xˉn using a

parametric bootstrap with B = 1000 bootstrap replications. As?sume that the samples come from a Normal distribution. You

should display a clearly labeled histogram of the sampling distri?bution of Xˉn. 3

(f) Using the parametric bootstrap replications above, estimate the

standard error of your estimate.

(g) Calculate approximate 95% confidence intervals for μ in two ways,

as described in part (d).

7. Estimating correlation

(a) Estimate the correlation ρ between midterm grades and home?work grades. The data may be downloaded from Canvas as is

called “hw3 prob7 dat.csv”. Use the command

data = read.csv(“hw3 prob7 dat.csv”) to read the data.

Use the sample correlation, ?ρ for which the R command is cor(·, ·).

(b) Use the nonparametric bootstrap with B = 1000 replicates to

estimate the sampling distribution of ?ρ. You should display a

clearly labeled histogram as well as report the bootstrap estimate

of the standard error of ?ρ.

(c) What is your conclusion based on your work above? Please be

as detailed as possible. (For instance, is there a relationship?

What kind of relationship? Is the relationship significant? How

strong/weak?)



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