Addendum to `Asymptotic inference for a
linear stochastic differential equation with
time delay'
ALEXANDER A. GUSHCHIN
1
and UWE KU
È
CHLER
2
1
Steklov Mathematical Institute, Gubkina 8, 117966 Moscow GSP-1, Russia.
E-mail: gushchin@mi.ras.ru
2
Institut fu
È
r Mathematik, Humboldt-Universita
È
t zu Berlin, Unter den Linden 6, D-10099 Berlin,
Germany. E-mail: kuechler@mathematik.hu-berlin.de
We strengthen the convergence result proving the local asymptotic mixed normality property in one of
the 11 cases considered in our previous paper.
Keywords: local asymptotic mixed normality; maximum likelihood estimator; stochastic differential
equations; time delay
In Gushchin and Ku
È
chler (1999) we considered the stochastic differential equation
dX (t) aX(t)dt bX (t ÿ 1) dt dW (t), t > 0, (1)
with initial condition X (t) X
0
(t), t 2 [ÿ1, 0], where (X
0
(t), t 2 [ÿ1, 0]) is a given
continuous stochastic process independent of a standard Wiener process (W (t), t > 0). The
parameter W (a, b)
is unknown and supposed to be estimated based on (X (t), t < T).
(The sign
denotes matrix or vector transposition.) The solutions (X (t), t 2 [ÿ1, T]) of the
differential equation (1) generate a family (P
W
T
, W 2 R
2
) of distributions on C([ÿ1, T ]). We
studied local asymptotic properties of this family and corresponding properties of the
maximum likelihood estimators of W for T !1.
In particular, for a ®xed W
0
(a, b)
2 R
2
the limit behaviour of
dP
W
T
dP
W
0
T
(X ), W 2 R
2
,
was studied. It turned out that this behaviour depends strongly on the position of W
0
, and one
has to consider 11 different cases (see Gushchin and Ku
È
chler 1999, Table 1, p. 1064). The
family (P
W
T
) is locally asymptotically normal in one of these cases, it is locally asymptotically
mixed normal (LAMN) in another three cases, and it is only locally asymptotically quadratic
in ®ve of the remaining cases. As for the last cases, called P1 and P2, we showed that the
family (P
W
T
) is still locally asymptotically quadratic but has an additional property that we
called periodic local asymptotic mixed normality (PLAMN).
Assume that we are in case P2, i.e. a , 1 and b , î=sin î, where î is the unique number
in (0, ð) with î a tan î (î ð=2ifa 0), or a > 1 and b , ÿe
aÿ1
. We refer to
Bernoulli 7(4), 2001, 629±632
1350±7265 # 2001 ISI/BS
Gushchin and Ku
È
chler (1999) for the de®nition of the random vector V
T
, the random matrix
I
T
, and the numbers v
0
. 0, î
0
. 0, A
0
and B
0
. It was shown that the following properties
hold (Gushchin and Ku
È
chler 1999, Proposition 2.6):
(i)
log
dP
W
0
e
ÿv
0
T
ì
T
dP
W
0
T
(X ) ì
V
T
ÿ
1
2
ì
I
T
ì, ì 2 R
2
; (2)
(ii) for T
n
u nÄ, where u 2 [0, Ä)is®xed,Ä ð=î
0
, n > 0,
(V
T
n
, I
T
n
) !
d
(V
1
(u), I
1
(u)), n !1,
where (V
1
(u), I
1
(u))
d
(I
1=2
1
(u) Z, I
1
(u)) and Z is a random vector independent of
I
1
(u) and distributed as N (0, I
2
). The matrix I
1
(u) is given by
I
1
(u)
1
0
e
ÿ2v
0
t
U
2
0
(u ÿ t)dt
1
0
e
ÿ2v
0
t
U
0
(u ÿ t)U
2
(u ÿ t)dt
1
0
e
ÿ2v
0
t
U
0
(u ÿ t)U
2
(u ÿ t)dt
1
0
e
ÿ2v
0
t
U
2
2
(u ÿ t)dt
0
B
B
B
@
1
C
C
C
A
, (3)
where
U
i
(t) X
0
(0)ö
i
(t) b
0
ÿ1
ö
i
(t ÿ s ÿ 1)e
ÿv
0
(s1)
X
0
(s)ds
1
0
ö
i
(t ÿ s)e
ÿv
0
s
dW (s), (4)
in which
ö
i
(t) A
i
cos(î
0
t) B
i
sin(î
0
t), i 0, 2 (5)
and
A
2
B
2
e
ÿv
0
cos î
0
ÿsin î
0
sin î
0
cos î
0
A
0
B
0
: (6)
We have referred to this asymptotic behaviour as PLAMN to indicate that the cluster points
of (V
T
, I
T
) have the same structure as in the LAMN case but that (V
T
, I
T
) converges in
distribution only if T runs to in®nity through a grid in such a way that the fractional part of
T=Ä tends to a limit.
The purpose of this note is to show that in case P2 one can prove the usual LAMN
property. To this end, we look for matrices Ö
T
, T > 0, such that
Ö
T
U
0
(T ÿ t)
U
2
(T ÿ t)
does not depend on T . In view of (4) and (5), this will be the case if
Ö
T
A
0
A
2
B
0
B
2
cos(T ÿ t)
sin(T ÿ t)
630 A.A. Gushchin and U. Ku
È
chler
is independent of T . Using (6), after elementary calculations one obtains that the latter
property holds if and only if
Ö
T
Ö
0
cos(î
0
T) ÿ sin(î
0
T) cot î
0
ÿe
ÿv
0
sin(î
0
T) csc î
0
e
v
0
sin(î
0
T) csc î
0
cos(î
0
T) sin(î
0
T) cot î
0
: (7)
De®ne Ö
T
according to (7) with Ö
0
: I
2
, where I
2
is the identity matrix. Then
~
U
0
(t)
~
U
2
(t)
!
: Ö
T
U
0
(T ÿ t)
U
2
(T ÿ t)
!
X
0
(0)
A
0
cos(î
0
t) ÿ B
0
sin(î
0
t)
e
ÿv
0
(A
0
cos(î
0
(t 1)) ÿ B
0
sin(î
0
(t 1)))
!
b
0
ÿ1
A
0
cos(î
0
(t s 1)) ÿ B
0
sin(î
0
(t s 1))
e
ÿv
0
(A
0
cos(î
0
(t s 2)) ÿ B
0
sin(î
0
(t s 2)))
!
e
ÿv
0
(s1)
X
0
(s)ds
1
0
A
0
cos(î
0
(t s)) ÿ B
0
sin(î
0
(t s))
e
ÿv
0
(A
0
cos(î
0
(t s 1)) ÿ B
0
sin(î
0
(t s 1)))
!
e
ÿv
0
s
dW (s):
Extend the de®nition of I
1
(u)tou 2 R
by the same formula (3). Then
Ö
T
I
1
(T)Ö
T
1
0
e
ÿ2v
0
t
Ö
T
U
0
(T ÿ t)
U
2
(T ÿ t)
U
0
(T ÿ t)
U
2
(T ÿ t)
Ö
T
dt,
and, therefore, the matrix
~
I
1
: Ö
T
I
1
(T)Ö
T
does not depend on T and is given by
~
I
1
1
0
e
ÿ2v
0
t
~
U
2
0
(t)dt
1
0
e
ÿ2v
0
t
~
U
0
(t)
~
U
2
(t)dt
1
0
e
ÿ2v
0
t
~
U
0
(t)
~
U
2
(t)dt
1
0
e
ÿ2v
0
t
~
U
2
2
(t)dt
0
B
B
B
@
1
C
C
C
A
:
Put
~
V
T
: Ö
T
V
T
,
~
I
T
: Ö
T
I
T
Ö
T
:
Proposition 1. In case P2 the family (P
W
, W 2 R
2
) is LAMN at every W
0
:
log
dP
W
0
e
ÿv
0
T
Ö
T
ì
T
dP
W
0
T
(X ) ì
~
V
T
ÿ
1
2
ì
~
I
T
ì (8)
and
(
~
V
T
,
~
I
T
) !
d
(
~
V
1
,
~
I
1
), (9)
where (
~
V
1
,
~
I
1
)
d
(
~
I
1=2
1
Z,
~
I
1
) and Z is a vector independent of
~
I
1
and distributed as
N (0, I
2
).
Proof. Equation (8) is obvious from (2).
Addendum to `Asymptotic inference for a linear SDE with time delay' 631
It was shown in the proof of Proposition 2.6 that
I
T
ÿ I
1
(T ) !
P
0:
Thus
~
I
T
!
P
~
I
1
because the matrices Ö
T
are uniformly bounded. Now the joint convergence (9) follows from
the stable limit theorem for martingales as in the proofs of Propositions 2.2±2.4 in Gushchin
and Ku
È
chler (1999). h
Remark. Contrary to the case just considered, in case P1 one cannot avoid periodic behaviour
of the likelihood and prove the LAMN property by choosing an appropriate matrix norming.
For a general statement of this type for delay processes, see Putschke (2000).
Acknowledgements
The authors express their thanks to anonymous referees, whose comments led to a more
detailed explanation of how to construct the matrices Ö
T
.
References
Gushchin, A.A. and Ku
È
chler, U. (1999) Asymptotic inference for a linear stochastic differential
equation with time delay. Bernoulli, 5, 1059±1098.
Putschke, U. (2000) Af®ne stochastische Funktionaldifferentialgleichungen und lokal asymptotische
Eigenschaften ihrer ParameterschaÈtzungen. Doctoral dissertation, Humboldt-UniversitaÈt zu Berlin.
Received June 2000 and revised March 2001
632 A.A. Gushchin and U. Ku
È
chler