MATH 758 – Computational Statistics - Homework on variance reduction
1. Suppose one observes independent random variables
p
XX ,,
1
K
, where
i
X
is
distributed
)1,(
i
N
. Suppose we judge the goodness of an estimator
)
ˆ
,,
ˆ
(
ˆ
1p
K=
by
the “sum of squared errors” loss function
=2
)
ˆ
(),
ˆ
(ii
L
and you wish to compute
the difference in risks
),
ˆ
(),
ˆ
(
0
JS
RR
, where
),,(
ˆ
10 p
XX K=
is the usual estimate
and
JS
ˆ
is the famous James-Stein estimate where the estimate of
i
is given by
i
j
X
X
c+
2
1
where 0 < c < 2 (p-2). (Above
()
+
a
means take the positive part of a.)
Focus on the case where p = 5 and c = 3, and focus on the computation of the difference
in risks when
5.1,0.1,5.,0
1
===
p
L
.
Compute the difference in risks by naïve Monte Carlo and by use of two variance
reduction methods. Base each computation on m = 1000 simulated samples. Explain the
variance reductions that you use. Demonstrate that you have achieved variance reduction
in each case.
Some sample code to compute the risk of the James-Stein estimate via naïve Monte Carlo
simulation is given by jamestein.R that is available in the montecarlo folder in the class
web site.
2. Exercise 6.6 (page 180) from book.