
Testing a parametric hypothesis 131
θ is one-dimensional and its estimator
b
θ is chosen as the root of the
equation
Z
e
T
(n)
0
h(s,θ)dM
n
(s,θ) = 0,
which is the maximum likelihood equation, then for the process
ξ
n
(t) =
1
(
R
τ
0
h
2
(s,
ˆ
θ)A
n
(ds,
ˆ
θ))
1/2
Z
t
0
h(s,
b
θ)M
n
(ds,
b
θ)
the limit statement
ξ
n
d
→V = v ◦F,
with
F(t) =
Z
t
0
h
2
(s,θ)A(ds, θ)
Z
∞
0
h
2
(s,θ)A(ds, θ) ,
is true.
Therefore, one can derive statistical inference about F in exactly
the same way as we have seen it in Lecture 5.
What happens if both parameters are unknown and have to be es-
timated? As in the general case, we start with linear regression of
dM
n
(t,
b
θ) on
Z
τ
t
h(s,θ)dM
n
(s,θ) =
1
θ
1
[M
n
(τ,θ) −M
n
(t,θ)]
R
τ
t
(
1
θ
2
+ s)d