Suppose that { y t : t ∈ Z } is a discrete-time stochastic process and we have observed the values y 1 y m − 1 , y m +1 , y n but y m has not been observed. Let f be the joint density of y 1 y n and...

Suppose that
{
y
t
:
t

∈

Z}
is a discrete-time stochastic process and we

have observed the values
y1y
m
−1, y
m+1, y
n
but
y
m
has not been observed.

Let
f
be the joint density of
y1y
n
and let
g
be the unimodal

conditional density of
y
m
given the observed values. Assume that the mode

and the mean of
g
coincide. Let
z
be the value that maximizes
f
with respect

to
ymis. Show that under these circumstances,
z
=
E[y
m
|
y1, y
m
−1, y
m+1y
n].