Assignment #5
STAT 992
Spring 2015
Complete the following problems below. Within each part, include your R program output with code
inside of it and any additional information needed to explain your answer. Your R code and output
should be formatted in the exact same manner as in the lecture notes.
1) (19 total points) The Rayleigh distribution is often used to help model wind speed (Celik, Energy
Conversion and Management, 2004, p. 1735-1747). Let X be a random variable denoting the
average wind speed during a February day in Lincoln, where X has a Rayleigh distribution defined
as
 1  21( x/ )2
 xe
f(x)   2

0

for x  0
for x  0
where  > 0. The data file wind_speed.csv contains the observed daily average wind speed values
for 2000 – 2004 in Lincoln during each February. Assume that each observation is independent
(the autocorrelations are all nonsignificant, except for the first autocorrelation in 2002).
a) Perform the derivations below without using a computer! All answers should be given in terms
of symbols rather than numerical values that use the observed data.
i) (2 points) Derive the maximum likelihood estimator of .
ii) (1 point) Derive the method of moment estimator for . Note that E(X) =   / 2 .
iii) (2 points) Through using the asymptotic normality of maximum likelihood estimators, find
the estimated asymptotic variance of the maximum likelihood estimator. Note that E(X) =
22.
b) (2 points) Find the numerical values for the method of moment estimate, the maximum
likelihood estimate, and the estimated asymptotic variance.
c) (2 points) Plot the log-likelihood function and first derivative of the log-likelihood function.
Include a line on each plot that shows the location of the maximum likelihood estimate relative
to the function.
d) The purpose of this problem to calculate some of the derivatives in part a) numerically using
fderiv() of the pracma package with the function’s defaults.
i) (3 points) Find the first derivative of the log-likelihood function using fderiv(). Evaluate
this numerical derivative at  equal to 7, the maximum likelihood estimate, and 8. Compare
these values with the actual values produced by the results in part b).
ii) (3 points) Find the second derivative of the log-likelihood function using fderiv().
Evaluate this numerical derivative at the maximum likelihood estimate and then use it to
compute the estimated asymptotic variance of the maximum likelihood estimator. Compare
the estimated variance to that produced by the results in part b).
e) (4 points) Through using optim() (method = "BFGS") with the data, find the maximum
likelihood estimate for  and the corresponding estimated asymptotic variance. Verify your
answers match the results given in part b). Use the method of moment estimate as the initial
value for .
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2) (9 total points) Continuing 1), the Rayleigh distribution is a special case of the Weibull distribution.
One representation of the Weibull PDF is
 1   x 
  


 x
f(x)    e

0

for x  0
for x  0
where  > 0 and  > 0. Note that when  = 2 and    2 , the Weibull simplifies to the Rayleigh
distribution. Complete the following.
a) (3 points) Find the method of moment estimates for  and  using the data. Note that
2
  2
 1 
2
2
2
E(X)  (1  1/  ) , E(X )   (1  2 /  ) , and Var(X)     1      1    . Show all

   
 
derivations.
b) (3 points) Find the maximum likelihood estimates for  and  using optim() and the observed
data. State the estimated covariance matrix.
c) (3 points) Construct contour and 3D plots of the log-likelihood function with the maximum
likelihood estimates plotted at their appropriate locations.
3) (3 points) Continuing 1) and 2), construct histograms and EDF plots of the data to examine
whether a Rayleigh and/or Weibull distribution work well for the data set. Comment on which
distribution is better.
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