Stat 5102 (Geyer) Spring 2012
Homework Assignment 7
Due Wednesday, March 21, 2012
Solve each problem. Explain your reasoning. No credit for answers with
no explanation. If the problem is a proof, then you need words as well as
formulas. Explain why your formulas follow one from another.
71.
Suppose
X
1
,
...
,
X
n
are IID Cauchy(
μ,σ
) and
σ
= 1 is known. We
wish to do maximum likelihood estimation, which cannot be done in closed
form, so you must use R. One needs a good estimate of the location param
eter to use as a starting point for the optimization. The location parameter
is the center of symmetry and also the median. Thus the sample median is
a good starting point. Data for the problem are at the URL
http://www.stat.umn.edu/geyer/5102/data/prob71.txt
(a) Find the MLE for these data.
(b) Find the observed Fisher information evaluated at the MLE.
(c) Find an asymptotic 95% conﬁdence interval for the parameter
μ
.
72.
Suppose
x
1
,
...
,
x
n
are known numbers (not random), and we ob
serve random variables
Y
1
,
...
,
Y
n
that are independent but
not
identically
distributed random variables having distributions
Y
i
∼ N
(
α
+
βx
i
,σ
2
)
,
where
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 Spring '03
 Staff
 Normal Distribution, Maximum likelihood, good starting point, mle, log likelihood, Fisher Information

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