Description
1. Let f(x) = (cos x)
2
for 0 < x < 2π.
(a) Graph the function f(x).
(b) Use Monte Carlo integration with uniform[0, 2π] sample size 1,000,000 to find the area
under f(x) on the range 0 < x < 2π, and to find a 95% confidence interval for the area.
(c) Use trigonometry or calculus to find the same area exactly. Did the confidence interval
cover the true value?
2. In this question, you will do some resampling and show results in graphics. This is related to
bootstrap technique. The population distribution is a normal with µ = 10 and σ
2 = 4. The
statistic is the sample mean. Hence in theory we know exactly what the density function of
the sample mean is.
(a) Simulate a sample, say x, with sample size n=100. Report its mean, sd, min, and max.
(b) Use R functions sample and replicate to resample x 50000 times with replacement. The
statistic is the sample mean and the output is booted.data. Find the mean, sd, min, and
max of booted.data.
(c) Plot the histogram of booted.data. Please double the cells of histogram since the default
one is too small. Please plot as a density plot since the theoretical density will be added
in the next step. Comment the shape and center of this distribution.
(d) Plot the histogram of booted.data-mean(x) with twice number of default cells. Please plot
as a density plot. Add the theoretical density function of the X¯ −µ to the histogram with
different line type and color. Comment out your findings.
(e) Repeat the procedures from (a) to (d) two additional times to check consistency.
3. For an odd number of data x = (x1, . . . , xn), the minimizer of the object function
f(θ|x) = Xn
i=1
|xi − θ|
is the sample median of x.
(a) Build an object function my.obj with arguments theta and x. The return value is
Pn
i=1 |xi − theta|. Try to vectorize your codes without any looping.
(b) Use the R function optimize to find the min value of the object function f(θ|x) for a given
data x. Please implement it as an R function with arguments x and interval=(min(x),
max(x)). The return value is the theta that minimizes the f. Test your function with the
dataset 3, 7, 9, 12, 15, 18, 21.
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(c) Use the R function nlminb to find the min value of the object function f(θ|x) for a given
data x. Please implement it as an R function with arguments x and start=mean(x). The
return value is the theta that minimizes the f. Test your function with the dataset 3, 7,
9, 12, 15, 18, 21.
(d) Test your functions from (b) and (c) with the dataset 1, 3, 7, 9, 12, 15, 18, 21. With
the function from (b), use three different wider intervals. With the function from (c),
use three different start values. What are your findings? For such a dataset, plot f with
theta from min(x) and max(x) and conclude your findings.
4. In this question you will use the R function nlminb to fit the Huron water levels with AR(1)
time series model. The time series huron records the mean water levels from 1860 to 1986.
Use the following steps to carry out your study.
(a) Open the dataset huron.R in owl and copy/paste into R to create an R object huron.
Compute its mean, sd, min, max, and median. Do boxplot and time series plot and
comment your findings.
(b) Let huron2=huron-mean(huron). Use nlminb to find the MLE of par[1] with the following
negative log likelihood function
log.likelihood=function(par, x){
n=length(x)
v=x[1]^2
for (i in 2:n)
v=v+(x[i]-par[1]*x[i-1])^2
return(v/par[2]+n*log(par[2]))
}
Make sure to choose a proper start value and the ranges that par[1] and par[2] within.
(c) Based on the estimator par[1] obtained in (b), compute the predicted values of huron2
as
pred.huron2=huron2
for (i in 2:length(huron2))
pred.huron2[i] = par[1]* huron2[i-1]
Then do a time series plot on huron2 and add pred.huron2 into it with different color.
Comment your findings.
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