.
Consider the model given byX
∼ Lognormal(0,
1) and log
Y
= 9 + 3 logX
+ , where
∼
N(0,
1). We wish to estimate
E{Y/X}. Compare the performance of the standard Monte Carlo estimator and the Rao–Blackwellized estimator.
.
Consider a bug starting at the origin of an infinite lattice
L
and walking one unit north, south, east, or west at each discrete time
t. The bug cannot stay still at any step. Let
x1:t
denote the sequence of coordinates (i.e., the path) of the bug up to time
t, say {x
i
= (vi,wi) :
i
= 1, . . . , t} with
x0 = (0,
0). Let the probability distribution for the bug’s path through time
t
be denoted
ft
(x1:t
) =
f1(x1)f2(x2|x1), . . . , ft
(x
t
|x1:t−1). Define
Dt
(x1:t
) to be the Manhattan distance of the bug from the origin at timet, namely
Dt
(x1:t
) = |vt| + |wt
|. Let
Rt
(v,w) denote the number of times the bug has visited the lattice point (v,w) up to and including time
t. Thus
Rt
(x
t
) counts the number of visits to the current location. The bug’s path is random, but the probabilities associated with moving to the adjacent locations at time
t
are not equal. The bug prefers to stay close to home (i.e., near the origin), but has an aversion to revisiting its previous locations. These preferences are expressed by the path distribution
ft
(x1:t
) ∝ exp{−(Dt
(x
t
) +
Rt
(x
t
)/2)}.
a.
Suppose we are interested in the marginal distributions of
Dt
(x
t
) and
Mt
(x1:t
) = max(v,w)∈L{Rt
(v,w)} where the latter quantity is the greatest frequency with which any lattice point has been visited. Use sequential importance sampling to simulate from the marginal distributions of
Dt
andMt
at time
t
= 30. Let the proposal distribution or envelope
gt
(x
t
|x1:t−1) be uniform over the four lattice points surrounding
x
t−1. Estimate the mean and standard deviation of
D30(x30) and
M30(x1:30).
b.
Let
gt
(x
t
|x1:t−1) be proportional to
ft
(x1:t) if
x
t
is adjacent to
x
t−1 and zero otherwise. Repeat part (a) using this choice for
gt
and discuss any problems encountered. In particular, consider the situation when the bug occupies an attractive location but
arrived there via an implausible path.
c.
A
self-avoiding walk
(SAW) is similar to the bug’s behavior above except that the bug will never revisit a site it has previously occupied. Simulation of SAWs has been important, for example, in the study of long-chain polymers [303, 394, 553]. Let
ft
(x1:t
) be the uniform distribution on all SAWs of length
t. Show that by using
gt
(x
t
|x1:t−1) =
ft
(x
t
|x1:t−1), the sequential update specified by
wt
=
wt−1ut
is given by
ut
=
ct−1 where
ct−1 is the number of unvisited neighboring lattice points
adjacent to
x
t−1 at time
t
− 1. Estimate the mean and standard deviation of
D30(x30) and
M30(x1:30). Discuss the possibility that the bug becomes entrapped.
d.
Finally, try applying the simplistic method of generating SAWs by simulating paths disregarding the self-avoidance requirement and then eliminating any selfinter secting paths post hoc. Compare the efficiency of this method to the approach in part (c) and how it depends on the total number of steps taken (i.e.,
t
_ 30).