Asymptotic theory of semiparametric Z-estimators for stochastic ...

The Annals of Statistics
2009, Vol. 37, No. 6A, 3555–3579
DOI: 10.1214/09-AOS693
© Institute of Mathematical Statistics, 2009
ASYMPTOTIC THEORY OF SEMIPARAMETRIC Z-ESTIMATORS
FOR STOCHASTIC PROCESSES WITH APPLICATIONS TO
ERGODIC DIFFUSIONS AND TIME SERIES
BY YOICHI NISHIYAMA
Institute of Statistical Mathematics
This paper generalizes a part of the theory of Z-estimation which has
been developed mainly in the context of modern empirical processes to the
case of stochastic processes, typically, semimartingales. We present a general
theorem to derive the asymptotic behavior of the solution to an estimating
equation θ n (θ, ĥn) = 0 with an abstract nuisance parameter h when the
compensator of n is random. As its application, we consider the estimation
problem in an ergodic diffusion process model where the drift coefficient
contains an unknown, finite-dimensional parameter θ and the diffusion coefficient
is indexed by a nuisance parameter h from an infinite-dimensional
space. An example for the nuisance parameter space is a class of smooth functions.
We establish the asymptotic normality and efficiency of a Z-estimator
for the drift coefficient. As another application, we present a similar result
also in an ergodic time series model.
1. Introduction. Let us begin with stating our motivating example; the details
are presented in Section 4. Consider the one-dimensional ergodic diffusion process
X on I = (l, r) ⊆ R which is a solution to the stochastic differential equation
(SDE) given by
∫ t
∫ t
(1)
X t = X 0 + S(X s ; θ)ds + σ(X s ; h) dW s ,
0
0
where s W s is a standard Brownian motion. Here, we consider a d-dimensional
parametric family {S(·; θ); θ ∈ } for the drift coefficient indexed by a compact
subset of R d , and a possibly infinite-dimensional “parametric” family
{σ 2 (·; h); h ∈ H } for the diffusion coefficient indexed by a (general) totally
bounded metric space (H, d H ). We denote by (θ 0 ,h 0 ) the true value of (θ, h).Our
aim is to estimate θ 0 when the model is perturbed by the unknown nuisance parameter
h. As for the parameter h 0 , we construct a d H -consistent estimator ĥn.
We prove that the Z-estimator ̂θ n , which is a solution to an estimating equation
n (θ, ĥn) = 0, is asymptotically normal and efficient. [We follow van der Vaart
and Wellner (1996) for the terminology “Z-estimator.”]
Received January 2009.
AMS 2000 subject classifications. Primary 62F12; secondary 62M05, 62M10.
Key words and phrases. Estimating function, nuisance parameter, metric entropy, asymptotic efficiency,
ergodic diffusion, discrete observation.
3555

3556 Y. NISHIYAMA
There exist a lot of works which treat the estimation problem for the drift coefficient.
It is well known that when the process X = (X t ) t∈[0,∞) is observed continuously
on the time interval [0,T], the diffusion coefficient may be assumed to
be known without loss of generality. (So, we may put h = h 0 .) In such cases, the
asymptotic normality and efficiency of the maximum likelihood estimator (MLE)
̂θ T for θ 0 ,asT →∞, has been already established. See, for example, Kutoyants
(2004). The MLE ̂θ T is a solution to the estimating equation ˙l T (θ) = 0 with
˙l T (θ) = 1 T
∫ T
0
Ṡ(X t ; θ)
σ 2 (X t ; h 0 ) [dX t − S(X t ; θ)dt],
where Ṡ denotes the derivative of S with respect to θ. On the other hand, when the
process X = (X t ) t∈[0,∞) is observed only at discrete time points {0 = t0 n

SEMIPARAMETRIC Z-ESTIMATION 3557
and to show the differentiability of (θ, h) ˜ n (θ, h) around (θ 0 ,h 0 ). Roughly
speaking, our main result asserts that if we assume h ↦→ σ 2 (·; h) is Lipschitz continuous
with respect to d H , that the metric entropy condition is satisfied,
∫ 1
0
√
log N(H,d H ,ε)dε

3558 Y. NISHIYAMA
denote by C ρ (T ) the space of functions on T which are continuous with respect to
the metric ρ. We equip both spaces with the uniform metric. Given a probability
measure P , we denote by P ∗ the corresponding outer probability; see van der Vaart
and Wellner (1996) for the stochastic convergence theory which does not assume
the measurability. We denote by p → and d → the convergence in (outer) probability
and the weak convergence. The limit notation mean in principle that we take the
limit as n →∞. The Euclidean metric on R d is denoted by ‖·‖.
2. General theory for semiparametric Z-estimation. Let two sets and H
be given. Let
n : × H → R d and ˜ n : × H → R d
be random maps. The latter should be a random “compensator” of the former,
and in the i.i.d. case it is not random and not depending on n. Compare the above
setting with that in Chapter 3.3 of van der Vaart and Wellner (1996)where˜ n ≡ .
We present a way to derive the asymptotic behaviour of the estimator ̂θ n for
the parameter θ ∈ of interest, with help of the estimator ĥn for the nuisance
parameter h ∈ H , which are solutions to the estimating equation
n (̂θ n , ĥn) ≈ 0.
Here, the true values θ 0 ∈ and h 0 ∈ H are supposed to satisfy
˜ n (θ 0 ,h 0 ) ≈ 0.
The following theorem extends a special case of Theorem 3.3.1 of van der Vaart
and Wellner (1996). See also Theorem 5.21 of van der Vaart (1998).
THEOREM 2.1. Let be a subset of R d with the Euclidean metric ‖·‖. Let
(H, d H ) be a semimetric space. Let n : × H → R d and ˜ n : × H → R d
be random maps defined on a probability space ( n , F n ,P n ).(We do not assume
any measurability.) Suppose that there exist a sequence of constants r n ↑∞, some
fixed point (θ 0 ,h 0 ) and an invertible matrix V θ0 ,h 0
which satisfy the following (i)
and (ii).
(i) There exists a neighborhood U ⊂ × H of (θ 0 ,h 0 ) such that
r n ( n − ˜ n ) d → Z
in l ∞ (U),
where almost all paths (θ, h) Z(θ,h) are continuous with respect to ρ =‖·‖∨
d H .
(ii) For given random sequence (̂θ n , ĥn), it holds that
˜ n (̂θ n , ĥn) − ˜ n (θ 0 ,h 0 ) − V θ0 ,h 0
(̂θ n − θ 0 ) = o P ∗ n
(r −1
n
+‖̂θ n − θ 0 ‖)

and that
SEMIPARAMETRIC Z-ESTIMATION 3559
‖̂θ n − θ 0 ‖∨d H (ĥn,h 0 ) = o P ∗ n
(1),
Then it holds that
˜ n (θ 0 ,h 0 ) = o P ∗ n
(r −1
n ).
r n (̂θ n − θ 0 ) d →−V −1
θ 0 ,h 0
Z(θ 0 ,h 0 ).
n (̂θ n , ĥn) = o P ∗ n
(r −1
n ),
To prove the above theorem, we need the following lemma, which is a slight
generalization of Lemma 19.24 of van der Vaart (1998).
LEMMA 2.2. Let (T , ρ) be a semimetric space. Suppose that Z n
d → Z in
l ∞ (T ) and that almost all paths of Z are continuous with respect to ρ. If T -valued
random sequence ̂t n satisfies ρ(̂t n ,t 0 ) = o P ∗ n
(1) for some nonrandom t 0 ∈ T , then
Z n (̂t n ) − Z n (t 0 ) = o P ∗ n
(1).
PROOF. Let us equip the space l ∞ (T ) × T with the metric ‖·‖ T + ρ, where
‖·‖ T denotes the uniform metric on l ∞ (T ). Define the function g : l ∞ (T ) × T →
R by g(z,t) = z(t) − z(t 0 ). Then for any z ∈ C ρ (T ) and t ∈ T , the function g
is continuous at (z, t). Indeed, if (z n ,t n ) → (z, t), then‖z n − z‖ T → 0, and thus
z n (t n ) = z(t n ) + o(1) → z(t), while z n (t 0 ) → z(t 0 ) is trivial.
By assumption, we have (Z n ,̂t n ) d → (Z, t 0 ) in l ∞ (T ) × T [see, e.g., Theorem
18.10(v) of van der Vaart (1998)]. Since almost all paths of Z belong to
C ρ (T ), by the continuous mapping theorem,
Z n (̂t n ) − Z n (t 0 ) = g(Z n ,̂t n ) d → g(Z,t 0 ) = Z(t 0 ) − Z(t 0 ) = 0.
The proof is finished.
□
PROOF OF THEOREM 2.1. Applying Lemma 2.2 to the l ∞ (U)-valued random
element Z n = r n ( n − ˜ n ),wehave
r n
(
n (̂θ n , ĥn) − ˜ n (̂θ n , ĥn) ) − r n
(
n (θ 0 ,h 0 ) − ˜ n (θ 0 ,h 0 ) ) = o P ∗ n
(1).
Since r n n (̂θ n , ĥn) = o P ∗ n
(1), wehave
r n
(˜ n (̂θ n , ĥn) − ˜ n (θ 0 ,h 0 ) ) =−r n n (θ 0 ,h 0 ) + o P ∗ n
(1).
By the assumption (ii), it holds that
(2) r n V θ0 ,h 0
(̂θ n − θ 0 ) =−r n n (θ 0 ,h 0 ) + o P ∗ n
(1 + r n ‖̂θ n − θ 0 ‖).
Now, since
r n ‖̂θ n − θ 0 ‖≤‖Vθ −1
0 ,h 0
‖r n ‖V θ0 ,h 0
(̂θ n − θ 0 )‖
≤ O Pn (1) + o P ∗ n
(1 + r n ‖̂θ n − θ 0 ‖) by (2),

3560 Y. NISHIYAMA
it holds that r n ‖̂θ n − θ 0 ‖=O P ∗ n
(1). Inserting this to (2), we have
r n V θ0 ,h 0
(̂θ n − θ 0 ) =−r n n (θ 0 ,h 0 ) + o P ∗ n
(1)
=−r n
(
n (θ 0 ,h 0 ) − ˜ n (θ 0 ,h 0 ) ) + o P ∗ n
(1)
d
→−Z(θ 0 ,h 0 ),
which implies the conclusion.
□
In Theorem 2.1, both “‖̂θ n − θ 0 ‖=o P ∗ n
(1)” and“d H (ĥn,h 0 ) = o P ∗ n
(1)” are
assumed. Under some conditions, the former automatically follows from the latter,
as it is seen in the following theorem.
THEOREM 2.3. Let (, d ) and (H, d H ) be two semimetric spaces. Let
n : × H → R d be a random map defined on a probability space ( n , F n ,P n ).
(We do not assume any measurability.) Let : × H → R d be a nonrandom
function. Suppose that
sup ‖ n (θ, h) − (θ,h)‖=o P ∗ n
(1).
(θ,h)∈×H
Suppose also that for some (θ 0 ,h 0 ) ∈ × H
inf ‖(θ,h 0)‖ > 0
θ : d (θ,θ 0 )>ε
∀ε>0,
and h (θ,h) is continuous at h 0 uniformly in θ. Then for any random sequence
(̂θ n , ĥn) such that n (̂θ n , ĥn) = o P ∗ n
(1) and that d H (ĥn,h 0 ) = o P ∗ n
(1), it
holds that d (̂θ n ,θ 0 ) = o P ∗ n
(1).
PROOF.
Observe that
‖(̂θ n , ĥn)‖≤‖(̂θ n , ĥn) − n (̂θ n , ĥn)‖+‖ n (̂θ n , ĥn)‖
≤ sup ‖(θ,h)− n (θ, h)‖+‖ n (̂θ n , ĥn)‖
θ,h
= o P ∗ n
(1).
Now, for every ε>0, there exist δ,η > 0 such that ‖(θ,h)‖ >ηfor every θ with
d (θ, θ 0 )>εand every h with d H (h, h 0 )ε} is
contained in the event {‖(̂θ n , ĥn)‖ >η}∪{d H (ĥn,h 0 ) ≥ δ}. The outer probability
of the latter event converges to 0. □
3. Uniform law of large numbers. In this section, we give a uniform law of
large numbers for ergodic processes, under a smoothness assumption. The proof is

SEMIPARAMETRIC Z-ESTIMATION 3561
standard, so it is omitted. [See, e.g., Theorem 2.4.1 of van der Vaart and Wellner
(1996) for the idea, or see Nishiyama (2009).]
THEOREM 3.1. Let (E, E) be a measurable space. Let beasetwhichis
totally bounded with respect to the semimetric ρ. Let a family {f(·; θ); θ ∈ }
of measurable functions on E be given. Suppose that there exists a measurable
function K such that
(3)
|f(x; θ)− f(x; θ ′ )|≤K(x)ρ(θ,θ ′ ) ∀θ,θ ′ ∈ .
(i) Suppose that the E-valued random process {X t } t∈[0,∞) is ergodic with the
invariant law μ, that is, for any μ-integrable function g
∫
1 T
∫
p
g(X t )dt → g(x)μ(dx).
T 0
E
If all f(·; θ) and K are μ-integrable, then
∫ 1 T
∫
sup
∣ f(X t ; θ)dt − f(x; θ)μ(dx)
T
∣ = o P ∗(1).
θ∈
0
(ii) Suppose that the E-valued random process {X i } i=1,2,... is ergodic with the
invariant law μ, that is, for any μ-integrable function g,
1
n
n∑
i=1
E
g(X i ) →
p ∫
g(x)μ(dx).
E
If all f(·; θ) and K are μ-integrable, then
1
n∑
∫
sup
f(X
∣
i ; θ)− f(x; θ)μ(dx)
n
E
∣ = o P ∗(1).
θ∈
i=1
REMARK. The smoothness assumption (3) can be replaced by “bracketing.”
See Theorem 2.4.1 of van der Vaart and Wellner (1996).
4. Ergodic diffusion processes.
4.1. Regularity conditions. Let us consider the diffusion process model introduced
in the first paragraph of Section 1. We shall list up some conditions.
We suppose that there exists a parametric family of d-dimensional vector-valued
functions {Ṡ(·; θ); θ ∈ } on I which satisfies the following conditions. Typically,
they may be considered to be the derivatives of S(·; θ) with respect to θ, thatis,
Ṡ(·; θ) = (
∂θ ∂
1
S(·; θ),...,
∂θ ∂
d
S(·; θ)) T . The function appearing in A1 and A3
may be chosen to be common without loss of generality.

3562 Y. NISHIYAMA
A1. is a compact subset of R d . There exists a measurable function on I
such that at the true θ 0 ∈ ,
S(x; θ)− S(x; θ 0 ) = Ṡ(x; θ 0 ) T (θ − θ 0 ) + (x)ɛ(x; θ,θ 0 ),
where sup x∈I |ɛ(x; θ,θ 0 )|=o(‖θ − θ 0 ‖) as θ → θ 0 .
A2. There exists a constant K>0 such that
sup |S(x; θ)− S(x ′ ; θ)|≤K|x − x ′ |;
θ∈
sup ‖Ṡ(x; θ)− Ṡ(x ′ ; θ)‖≤K|x − x ′ |;
θ∈
sup |σ 2 (x; h) − σ 2 (x ′ ; h)|≤K|x − x ′ |.
h∈H
A3. There exists a measurable function on I such that
sup |S(x; θ)|≤(x);
θ∈
sup ‖Ṡ(x; θ)‖≤(x);
θ∈
c := inf inf σ 2 (x; h) > 0;
h∈H x∈I
‖Ṡ(x; θ)− Ṡ(x; θ)‖≤(x)‖θ − θ ′ ‖ ∀θ,θ ′ ∈ ;
|σ 2 (x; h) − σ 2 (x; h ′ )|≤(x) d H (h, h ′ ) ∀h, h ′ ∈ H.
A4. sup t∈R E((X t ) 8 +|X t | 4 )ε
I
0
Ṡ(x; θ)
σ 2 (x; h 0 ) [S(x; θ 0) − S(x; θ)]μ(dx)
∥ > 0.
REMARK. The last assumption in A2 implies that σ 2 (x; h 0 ) ≤ C(1 +|x|) for
a constant C>0.

SEMIPARAMETRIC Z-ESTIMATION 3563
To close this subsection, let us discuss the possibility of the choice of the nuisance
parameter space (H, d H ).
EXAMPLE 1 (Parametric model). When (H, d H ) is a compact subset of a
finite-dimensional Euclidean space, the metric entropy condition A7 is indeed satisfied.
So the main restriction is the Lipschitz continuity of h ↦→ σ 2 (·; h) in A3.
This situation is more general than that in Yoshida (1992) and Kessler (1997),
although, as announced in Section 1, our result does not include theirs.
EXAMPLE 2 (The class of smooth functions). Let us consider the parametrization
σ(x; h) = h(x) where h is an element of the class H = CM α (I) defined
below. We equip the function space H with the uniform metric ‖·‖ ∞ for which
the last requirement in A3 is always fulfilled. To check A7, first we consider the
case where I is a bounded subset of R, and next we give some remarks for the
general case.
We take the material below from Section 2.7.1 of van der Vaart and Wellner
(1996). Let I be a bounded, convex subset of R q . (In the current example of onedimensional
diffusions, we are considering the case q = 1, but for the generality
we set q to be a general positive integer; see Section 5.) Let α>0andM>0
be given, and let α be the greatest integer smaller than α. For any vector k =
(k 1 ,...,k q ) of q integers, we define
D k =
∂x k ,
1
1 ···∂xk q
q
where k· = ∑ q
i=1 k i. We denote by CM α (I) the class of functions defined on I such
that
max sup
k·≤α x
|Dk h(x)|+max sup |D k h(x) − D k h(y)|
k·=α x,y ‖x − y‖ α−α ≤ M,
where the sumprema are taken over all x, y in the interior of I with x ≠ y. Then
there exists a constant K>0 depending only on α and q, such that
∂ k·
( M
log N(CM α (I), ‖·‖ ∞,ε)≤ Kλ(I 1 )
ε
) q/α
,
where λ(I 1 ) is the Lebesgue measure of the set {x : ‖x−I‖ < 1}. Hence, the metric
entropy condition A7 is satisfied if q/(2α) < 1, and therefore our theory works.
When I = R q , we shall restrict out attention, for example, to the following class
H of functions on R q .LetI 0 be a bounded, convex subset of R q , and we suppose
that the restriction of h ∈ H to I 0 belongs to C M (I 0 ) and that
(4)
sup |h(x) − h ′ (x)|≤L sup |h(x) − h ′ (x)| ∀h, h ′ ∈ H,
x∈R q x∈I 0

3564 Y. NISHIYAMA
for a constant L>0. Then both the last condition of A3 and A7 are satisfied. The
condition (4) is satisfied if we assume, for example, either of the following:
(i) h is known on I c 0 ;
(ii) when q = 1andI 0 =[l 0 ,r 0 ], each h is constant on (−∞,l 0 ] and on
[r 0 , ∞).
Although the examples (i) and (ii) might look restrictive, it should be noted that
in practice we can choose an arbitrary large I 0 .
Another way to deal with the unbounded case I = R q is to consider the parametrization
σ(x; h) = h(u(x)), h ∈ C α M (I 0),
where I 0 being a bounded, convex subset of R q′ and u : R q → I 0 is a fixed function.
If q ′ /(2α) < 1, then both the last requirement in A3 and the metric entropy
condition A7 are satisfied for the uniform metric d H =‖·‖ ∞ .
Instead of CM α (I), another possibility of the choice of H which satisfies the
metric entropy condition for the uniform metric is the Sobolev class; see Example
19.10 of van der Vaart (1998).
4.2. Results. As announced in Section 1, we propose to use the estimating
function
n (θ, h) = 1 n∑ Ṡ(X t n
i−1
; θ)
tn
σ 2 (X t n
i−1
; h) [X ti
n − X t n
i−1
− S(X t n
i−1
; θ)|ti n − ti−1 n |],
i=1
whose compensator is
˜ n (θ, h) = 1 n∑ Ṡ(X t n
i−1
; θ) [∫ t n
i
tn
σ 2 (X t n
i−1
; h)
i=1
t n i−1
Then we have the following two lemmas.
S(X t ,θ 0 )dt − S(X t n
i−1
; θ)|t n i − t n i−1 | ].
LEMMA 4.1. Assume n → 0 and tn n →∞. Equip the space × H with the
metric ρ =‖·‖∨d H . Under A1–A5 and A7, √ tn ( n − ˜ n ) converges weakly in
C ρ ( × H) to a zero-mean Gaussian process Z with the covariance
∫
EZ(θ, h)Z(θ ′ ,h ′ ) T =
I
Ṡ(x; θ)Ṡ(x; θ ′ ) T
σ 2 (x; h)σ 2 (x; h ′ ) σ 2 (x; h 0 )μ(dx).
In particular, the random variable Z(θ 0 ,h 0 ) is distributed with N (0,I(θ 0 ,h 0 )).
LEMMA 4.2. Assume n = o((tn n)−1 ) and tn n →∞. Under A1–A6, for any
random sequence (θ n ,h n ) such that ‖θ n − θ 0 ‖∨d H (h n ,h 0 ) = o P ∗(1), it holds that
˜ n (θ n ,h n ) − ˜ n (θ 0 ,h 0 ) − (−I(θ 0 ,h 0 ))(θ n − θ 0 ) = o P ∗( (t
n
n ) −1/2 +‖θ n − θ 0 ‖ ) .

SEMIPARAMETRIC Z-ESTIMATION 3565
Combining these lemmas with Theorem 2.1, and noting also ˜ n (θ 0 ,h 0 ) =
O P ( 1/2
n ) which will be proved by using Lemma 4.5 below, we can conclude
the following theorem.
THEOREM 4.3. Assume n = o((tn n)−1 ) and tn n →∞. Under A1–A7, for any
random sequence (̂θ n , ĥn), such that
and
‖̂θ n − θ 0 ‖=o P ∗(1),
d H (ĥn,h 0 ) = o P ∗(1)
n (̂θ n , ĥn) = o P ∗((t n n )−1/2 ),
the estimator ̂θ n is asymptotically normal and efficient:
√
t n n (̂θ n − θ 0 ) d → N (0,I(θ 0 ,h 0 ) −1 ).
When A8 is also satisfied, the assumption “‖̂θ n − θ 0 ‖=o P ∗(1)” is automatically
satisfied.
In the above theorem, the only assumption which we cannot check in the course
of computing the data is the consistency “d H (ĥn,h 0 ) = o P ∗(1),” because it involves
the true value h 0 of the unknown parameter h ∈ H .When{σ 2 (·; h); h ∈ H }
is a class of functions σ 2 (·; h) = h(·) where H is a class of smooth functions, one
may think that a kernel estimator is a candidate for ĥn. As stated above, in view of
the Lipschitz condition of h ↦→ σ 2 (·; h) (the last condition in A3), it is convenient
to consider the consistency with respect to the uniform metric. However, to show
the consistency of the kernel estimator with respect to the uniform metric is a task.
Generally speaking, showing the consistency for infinite-dimensional parameter
is not a trivial problem, which should be solved by independent articles. See, for
example, Hoffmann (2001). Below, we give a general way to show the consistency
of a least square estimator.
THEOREM 4.4.
Assume n → 0 and tn n →∞. Assume A2–A5 and A7. Suppose
that
∫
inf |σ 2 (x; h) − σ 2 (x; h 0 )| 2 μ(dx) > 0 ∀ε>0
h : d H (h,h 0 )>ε I
is satisfied. If the random element ĥn satisfies A n (ĥn) ≤ inf h∈H A n (h) + o P ∗(1)
where
A n (h) = 1
n∑
|X t n
i
− X t n
i−1
| 2
tn
∣
|ti n − ti−1 n | − σ 2 2
(X t n
i−1
,h)
∣ |ti n − ti−1 n |,
i=1
then it holds that d H (ĥn,h 0 ) = o P ∗(1).

3566 Y. NISHIYAMA
4.3. Proofs. Before the proofs, we state a lemma which is well known.
LEMMA 4.5.
|t n i
− t n i−1 |≤1.
Let X be a solution to the SDE (1) for (θ, h) = (θ 0 ,h 0 ). Assume
(i) For any k ≥ 2, there exists a constant C k > 0, depending only on k, such
that
E sup |X t − X t n
t∈[ti−1 n ,tn i ] i−1
| k ≤ C k sup E{|S(X s ; θ 0 )| k +|σ(X s ; h 0 )| k }|ti n − ti−1 n |k/2
s∈R
=: D k |t n i − t n i−1 |k/2 ,
provided the right-hand side is finite.
(ii) For any k ≥ 2 and any measurable function f , g, it holds that
sup E ( |X t − X t n
t∈[ti−1 n ,tn i ] i−1
| k/2 |f(X t n
i−1
)||g(X t )| )
≤ (D k |ti n − ti−1 n |k/2 ) 1/2 sup(E|f(X s )| 4 ) 1/4 sup(E|g(X s )| 4 ) 1/4 ,
s∈R
s∈R
provided the right-hand side is finite.
PROOF. The assertion (i) is well known. (Use Hölder’s inequality
∫ t n
and Burkholder–Davis–Gundy’s inequality for i
ti−1
n |S(X s ; θ 0 )| ds and
sup t∈[t n
i−1 ,ti n] | ∫ t
t n σ(X s ; h 0 )dW s |.) The assertion (ii) follows from Hölder’s inequality
and (i).
i−1
□
During the proofs, we write
ψ(x; θ,h)=
Ṡ(x; θ)
σ 2 (x; h) ,
which is a d-dimensional vector-valued function. For each component ψ (j) (x; θ,
h) (j = 1,...,d), it holds that
∣ ψ (j) (x; θ,h)− ψ (j) (x ′ ; θ,h) ∣ ∣
(5)
≤ |Ṡ(j) (x; θ)− Ṡ (j) (x ′ ; θ)|
σ 2 (x; h)
+ ∣ Ṡ (j) (x ′ ; θ) ∣ 1
∣
σ 2 (x; h) − 1
σ 2 (x ′ ; h)
∣
{ 1
≤
c + (x) }
c 2 K|x − x ′ |

and that
∣ ψ (j) (x; θ,h)− ψ (j) (x; θ ′ ,h ′ ) ∣ ∣
(6)
SEMIPARAMETRIC Z-ESTIMATION 3567
≤ |Ṡ(j) (x; θ)− Ṡ (j) (x; θ ′ )|
σ 2 (x; h)
+ ∣ ∣Ṡ (j) (x; θ ′ ) ∣ 1
∣
σ 2 (x; h) − 1
σ 2 (x; h ′ )
∣
≤ |Ṡ(j) (x; θ)− Ṡ (j) (x; θ ′ )|
+ ∣ c
Ṡ (j) (x; θ ′ ) ∣ |σ 2 (x; h) − σ 2 (x; h ′ )|
c 2
{ (x)
≤ + |(x)|2 } (‖θ − θ ′
c c 2 ‖∨d H (h, h ′ ) ) .
PROOF OF LEMMA 4.1. We apply Theorem 3.4.2 of Nishiyama (2000b) [or,
see Theorem 3.3 of van der Vaart and van Zanten (2005)] to the terminals M n,θ,h
tn
of the continuous martingales t Mt
n,θ,h given by
M n,θ,h
t = 1 √ t n n
n∑
∫ t n
i ∧t
ψ(X t n
i−1
; θ,h) σ(X s ; h 0 )dW s .
i=1
ti−1 n ∧t
For the finite-dimensional convergence, it is sufficient to show the convergence of
predictable covariation. This is done as follows.
(7)
〈M n,θ,h ,M n,θ ′ ,h ′ 〉 t n = 1 n∑
∫ t n
tn
ψ(X t n
i−1
; θ, h)ψ(X t n
i−1
; θ ′ ,h ′ ) T i
= 1
i=1
∫ t n n
t n i−1
σ 2 (X s ; h 0 )ds
ψ(X t ; θ, h)ψ(X t ; θ ′ ,h ′ ) T σ 2 (X t ; h 0 )dt + o P (1)
tn
0
∫
p
→ ψ(x; θ, h)ψ(x; θ ′ ,h ′ ) T σ 2 (x; h 0 )μ(dx)
I
∫
Ṡ(x; θ)Ṡ(x; θ ′ ) T
=
I σ 2 (x; h)σ 2 (x; h ′ ) σ 2 (x; h 0 )μ(dx)
= I(θ 0 ,h 0 ) if (θ, h) = (θ ′ ,h ′ ) = (θ 0 ,h 0 ).
Here, to show (7), we have used the bound (5) and Lemma 4.5(ii) twice.
To establish Nishiyama’s condition [ME], let us observe the following fact to
check the metric entropy condition for the product space × H .
In general, if (D, d) and (E, e) are two semimetric spaces, then the covering
number of the product space D × E with respect to the maximum semimetric
d ∨ e, namely N(D × E,d ∨ e,ε), is bounded by N(D,d,ε)· N(E,e,ε).Tosee
this claim, let B i , i = 1,...,N(D,d,ε) be an ε-covering of D, andletC j , j =
1,...,N(E,e,ε)be an ε-covering of E. Then the diameters of the sets B i × C j ⊂

3568 Y. NISHIYAMA
D × E with respect to d ∨ e are smaller than ε, thus these sets form an ε-covering
of D × E. The claim has been proved. Consequently, the metric entropy condition
is satisfied if
∫ 1
0
∫ 1
0
√
log N(D,d,ε)dε < ∞
√
log N(D × E,d ∨ e,ε)dε < ∞
and
∫ 1
0
√
log N(E,e,ε)dε

to (θ 0 ,h 0 ) that
(8)
1
t n n
∫ t n n
0
SEMIPARAMETRIC Z-ESTIMATION 3569
ψ(X t ; θ n ,h n )[S(X t ; θ 0 ) − S(X t ; θ n )] dt
= 1
t n n
= 1
=
tn
∫
I
∫ t n n
0
∫ t n n
0
ψ(X t ; θ n ,h n )Ṡ(X t ; θ 0 ) T dt(θ 0 − θ n ) + o P ∗(‖θ n − θ 0 ‖)
ψ(X t ; θ 0 ,h 0 )Ṡ(X t ; θ 0 ) T dt(θ 0 − θ n ) + o P ∗(‖θ n − θ 0 ‖)
ψ(x; θ 0 ,h 0 )Ṡ(x; θ 0 ) T μ(dx)(θ 0 − θ n ) + o P ∗(‖θ n − θ 0 ‖)
=−I(θ 0 ,h 0 )(θ n − θ 0 ) + o P ∗(‖θ n − θ 0 ‖).
To prove (8) in the above computation, use (6) to show that for every j,k =
1,...,d
∫ 1 t n n
∣ ψ (j) (X t ; θ n ,h n )Ṡ (k) (X t ; θ 0 )dt − 1 ∫ t n n
ψ (j) (X t ; θ 0 ,h 0 )Ṡ (k) (X t ; θ 0 )dt
∣
t n n
0
≤ 1 ∫ t n{ n (Xt )
tn
0 c
= O P (1) · o P ∗(1)
= o P ∗(1).
The proof is complete.
t n n
0
+ |(X t)| 2
c 2 } ∣∣Ṡ (k) (X t ; θ 0 ) ∣∣ dt ·‖θ n − θ 0 ‖∨d H (h n ,h 0 )
□
PROOF OF THEOREM 4.3. By Lemma 4.5, it is easy to see that ˜ n (θ 0 ,h 0 ) =
O P ( 1/2
n ) = o P ((tn n)−1/2 ). So the main assertion follows from Theorem 2.1
with help from Lemmas 4.1 and 4.2. On the other hand, since it follows from
Lemma 4.5(ii) and Theorem 3.1(i) that sup θ,h ‖ n (θ, h) − (θ,h)‖=o P ∗(1),
where
∫
Ṡ(x; θ)
(θ,h) =
I σ 2 (x; h) [S(x; θ 0) − S(x; θ)]μ(dx),
the assertion that the consistency “‖̂θ n − θ 0 ‖=o P ∗(1)” automatically follows
from A8 is immediate from Theorem 2.3. □
PROOF OF THEOREM 4.4.
n∑
M n (h) = 1
M(h) =
tn
i=1
∫
I
Put
|σ 2 (X t n
i−1
; h) − σ 2 (X t n
i−1
; h 0 )| 2 |t n i − t n i−1 |,
|σ 2 (x; h) − σ 2 (x; h 0 )| 2 μ(dx).

3570 Y. NISHIYAMA
Let us apply Corollary 3.2.3 of van der Vaart and Wellner (1996) to the above
M n and M for the given ĥn which is the solution of A n (ĥn) ≤ inf h∈H A n (h) +
o P ∗(1). By Lemma 4.5(ii) and Theorem 3.1(i), it is not difficult to see that
sup h∈H |M n (h) − M(h)| =o P ∗(1), so it is sufficient to show that M n (ĥn) =
o P ∗(1).
Observe that
1
n∑
|X t n
i
− X t n
i−1
| 2
tn
∣
|t n i=1 i
− ti−1 n | − σ 2 2
(X t n
i−1
; h)
∣ |ti n − ti−1 n |
= 1
n∑
|X t n
i
− X t n
i−1
| 2
tn
∣
|t n i=1 i
− ti−1 n | − σ 2 2
(X t n
i−1
; h 0 )
∣ |ti n − ti−1 n |
+ 2 n∑
( |Xt n
i
− X t n
i−1
| 2
)
tn
|t n i=1 i
− ti−1 n | − σ 2 (X t n
i−1
; h 0 )
+ 1
t n n
× ( σ 2 (X t n
i−1
; h 0 ) − σ 2 (X t n
i−1
; h) ) |t n i − t n i−1 |
n∑
|σ 2 (X t n
i−1
; h 0 ) − σ 2 (X t n
i−1
; h)| 2 |ti n − ti−1 n |.
i=1
Let us prove that the supremum with respect to h of the absolute value of the
second term on the right-hand side converges in outer probability to zero [say, the
claim (a)].
Since we have from Itô’s formula that
|X t n
i
− X t n
i−1
| 2 = 2
∫ t n
i
t n i−1
(X s − X t n
i−1
)S(X s ; θ 0 )ds
∫ t n ∫
i
t n
i
+ 2 (X s − X t
ti−1
n n
i−1
)σ (X s ; h 0 )dW s + σ 2 (X s ; h 0 )ds,
ti−1
n
it is sufficient to show that C 1,n = o P (1), sup h∈H |C 2,n (h)|=o P ∗(1) and C 3,n =
o P (1), where
C 1,n = 1
t n n
C 2,n (h) = 1
t n n
C 3,n = 1
t n n
n∑
∫ t n i
∣ (X s − X t
i=1
ti−1
n n
i−1
)S(X s ; θ 0 )ds
∣ (X ti−1 n ),
n∑
∫ t n
i
(
(X s − X t
i=1
ti−1
n n
i−1
)σ (X s ; h 0 )dW s σ 2 (X t n
i−1
; h 0 ) − σ 2 (X t n
i−1
; h) ) ,
n∑
∫ t n
i
|σ 2 (X s ; h 0 ) − σ 2 (X t
i=1
ti−1
n n
i−1
; h 0 )| ds(X t n
i−1
).
By using Lemma 4.5(ii), we easily have EC 1,n → 0andEC 3,n → 0. On the other
hand, by using Theorem 3.4.2 of Nishiyama (2000b), it holds that C 2,n converges

SEMIPARAMETRIC Z-ESTIMATION 3571
weakly to zero in C dH (H ) (recall the argument in the proof of Lemma 4.1). Therefore,
we have sup h∈H |C 2,n (h)|=o P ∗(1).
Hence, the claim (a) is true, and we have that A n (ĥn) ≤ inf h∈H A n (h) + o P ∗(1)
implies that M n (ĥn) = o P ∗(1). The proof is finished. □
5. Ergodic time series.
5.1. Model and regularity conditions. Let us consider the time series model
given by
X i = ˜S(X i−1 ,...,X i−q1 ; θ)+ ˜σ(X i−1 ,...,X i−q2 ; h)w i .
By putting q = q 1 ∨q 2 and changing the domains of the functions ˜S and ˜σ , without
loss of generality, we can write
X i = S(X i ; θ)+ σ(X i ; h)w i ,
where X i = (X i−1 ,...,X i−q ) and S(·; θ) and σ(·; h) are some measurable functions
on R q . For simplicity, we assume that the initial values (X 0 ,...,X 1−q ) =
(x 0 ,...,x 1−q ) are fixed.
As for the noise {w i }, we consider the following two cases:
CASE G (Gaussian).
N (0, 1).
CASE M (Martingale).
where F i = σ {X j : j ≤ i}.
{w i } are independently, identically distributed with
E[w i |F i−1 ]=0andE[w 2 i |F i−1] =1 almost surely,
Clearly, the Case G is a special case of the Case M. When we do not especially
declare the restriction to the Case G, we consider the Case M in principle.
Let us list up some conditions which have the same fashion as those in Section
4.1. We suppose that there exists a parametric family of d-dimensional vectorvalued
functions {Ṡ(·; θ); θ ∈ } on R q which satisfies the following conditions.
Typically, they may be considered to be the derivatives of S(·; θ) with respect to θ,
that is, Ṡ(·; θ) = (
∂θ ∂
1
S(·; θ),...,
∂θ ∂
d
S(·; θ)) T . The function appearing in B1
and B2 may be chosen to be common without loss of generality.
B1. is a compact subset of R d . There exists a measurable function on R q
such that at the true θ 0 ∈ ,
S(x; θ)− S(x; θ 0 ) = Ṡ(x; θ 0 ) T (θ − θ 0 ) + (x)ɛ(x; θ,θ 0 ),
where sup x∈R q |ɛ(x; θ,θ 0 )|=o(‖θ − θ 0 ‖) as θ → θ 0 .

3572 Y. NISHIYAMA
B2. There exists a measurable function on R q such that
sup |S(x; θ)|≤(x);
θ∈
sup ‖Ṡ(x; θ)‖≤(x);
θ∈
σ 2 (x; h 0 ) ≤ (x),
c := inf inf σ 2 (x; h) > 0;
h∈H x∈R q
‖Ṡ(x; θ)− Ṡ(x; θ)‖≤(x)‖θ − θ ′ ‖ ∀θ,θ ′ ∈ ;
|σ 2 (x; h) − σ 2 (x; h ′ )|≤(x)d H (h, h ′ ) ∀h, h ′ ∈ H.
B3. The process {X i } i=1,2,... is ergodic under the true (θ 0 ,h 0 ) in the sense that
for q ′ = q and q + 1 there exists the invariant measure μ q ′ such that for every
μ q ′-integrable function f
1
n
n∑
i=1
We also assume that
∫
B4. The matrix
f(X i−1 ,...,X i−q ′) →
p ∫
f(x 1,...,x q ′)μ q ′(dx 1 ···dx q ′).
R q′
∫
R q (x)5 μ q (dx)ε
∥
∫ 1
0
√
log N(H,d H ,ε)dε 0.
See the end of Section 4.1 for the discussion of the choice of (H, d H ).
5.2. Results. In order to explain the idea of our estimating function, let us first
consider the Case G. We denote by P n,u the distribution of {X 1 ,...,X n } under

SEMIPARAMETRIC Z-ESTIMATION 3573
θ = θ 0 + n −1/2 u and h = h 0 ,whereu ∈ R d . By an easy computation, the loglikelihood
ratio is given by
(9)
where
and
n,u =
log dP n,u
dP n,0
(X 1 ,...,X n )
=−
n∑
i=1
1
2σ 2 (X i ; h 0 )
= n,u − B n,u ,
n∑
i=1
B n,u =
×{|X i − S(X i ; θ 0 + n −1/2 u)| 2 −|X i − S(X i ; θ 0 )| 2 }
1
σ 2 (X i ; h 0 ) {X i − S(X i ; θ 0 )}{S(X i ; θ 0 + n −1/2 u) − S(X i ; θ 0 )}
n∑
i=1
1
2σ 2 (X i ; h 0 ) {S(X i; θ 0 + n −1/2 u) − S(X i ; θ 0 )} 2 .
Under the above regularity conditions, we have
n,u
d → N (0,u T I(θ 0 ,h 0 )u) and B n,u
p
→
1
2
u T I(θ 0 ,h 0 )u.
So it follows from the theory of the local asymptotic normality that the distribution
of the asymptotically efficient bound in the Case G when h 0 is known is
N (0,I(θ 0 ,h 0 ) −1 ). That is, if we obtain an estimator ˜θ n such that √ n(˜θ n − θ 0 ) →
d
N (0,I(θ 0 ,h 0 ) −1 ), it is asymptotically efficient in the sense of the local asymptotic
minimax theorem. [See, e.g., Chapter 3.11 of van der Vaart and Wellner (1996).]
If the parameter h is unknown, then the estimation problem for θ becomes more
difficult. So if we have an estimator which asymptotically behaves as stated above,
then we may say that it is asymptotically efficient with the nuisance parameter h.
This argument is not true in the Case M where the log-likelihood does not equal
the formula (9), but we propose to use it for deriving an estimating equation which
yields the same asymptotic distribution as the Case G.
Not only in the Case G but also in the case M, differentiating (9) formally,
and replacing the true h 0 by the unknown parameter h, we propose the estimating
function
n (θ, h) = 1 n∑ Ṡ(X i ; θ) (
Xi
n σ 2 − S(X i ; θ) ) .
(X i ; h)
Its compensator is
˜ n (θ, h) = 1 n
n∑
i=1
i=1
Ṡ(X i ; θ) ( S(Xi
σ 2 ; θ 0 ) − S(X i ; θ) ) .
(X i ; h)

3574 Y. NISHIYAMA
Thus, it holds that
√ ( n n (θ, h) − ˜ n (θ, h) ) = √ 1 n∑ Ṡ(X i ; θ)
n σ 2 (X i ; h) σ(X i; h 0 )w i ,
which is the summation of a C ρ ( × H)-valued martingale difference array where
ρ =‖·‖∨d H . By using Jain–Marcus’ central limit theorem for martingales given
by Nishiyama (1996, 2000a, 2000b), we have the following lemma which plays a
key role in our approach.
LEMMA 5.1. Under B1–B3 and B5, the sequence of random fields √ n( n −
˜ n ), with parameter (θ, h), converges weakly in C ρ ( × H) to a zero-mean
Gaussian random field Z with the covariance
EZ(θ, h)Z(θ ′ ,h ′ ) T Ṡ(x; θ)Ṡ(x; θ
=
∫R ′ ) T
q σ 2 (x; h)σ 2 (x; h ′ ) σ 2 (x; h 0 )μ q (dx).
In particular, the random variable Z(θ 0 ,h 0 ) is distributed with N (0,I(θ 0 ,h 0 )).
Another lemma which is necessary to apply Theorem 2.1 is the following.
LEMMA 5.2. Under B1–B4, for any random sequence (θ n ,h n ) such that
‖θ n − θ 0 ‖∨d H (h n ,h 0 ) = o P ∗(1), it holds that
˜ n (θ n ,h n ) − ˜ n (θ 0 ,h 0 ) − (−I(θ 0 ,h 0 ))(̂θ n − θ 0 ) = o P ∗(‖θ n − θ 0 ‖).
i=1
Noting also that ˜ n (θ 0 ,h 0 ) = 0, we can apply Theorem 2.1 to conclude the
following theorem.
THEOREM 5.3.
Under B1–B5, for any random sequence (̂θ n , ĥn) such that
‖̂θ n − θ 0 ‖=o P ∗(1), d H (ĥn,h 0 ) = o P ∗(1) and n (̂θ n , ĥn) = o P ∗(n −1/2 ),
the estimator ̂θ n is asymptotically normal:
√ n(̂θ n − θ 0 ) d → N (0,I(θ 0 ,h 0 ) −1 ).
In particular, in the Case G, the estimator ̂θ n is asymptotically efficient. When B6
is also satisfied, the assumption “‖̂θ n − θ 0 ‖=o P ∗(1)” is automatically satisfied.
By the same reason as in Section 4, it is necessary to develop a procedure to construct
a consistent estimator ĥn for the nuisance parameter h ∈ H . The following
theorem gives us an answer.

SEMIPARAMETRIC Z-ESTIMATION 3575
THEOREM 5.4. Assume B2, B3 and B5.
(First step: initial estimator for θ 0 .) Suppose the identifiability condition
∫
inf |S(x; θ)− S(x; θ 0)| 2 μ q (dx)>0 ∀ε>0
θ : ‖θ−θ 0 ‖>ε R q
is satisfied. If a random sequence θn
LS satisfies A n (θn
LS)
≤ inf θ∈ A n (θ) + o P ∗(1),
where
A n (θ) = 1 n∑
|X i − S(X i ; θ)| 2 ,
n
i=1
then it holds that ‖θn
LS − θ 0 ‖=o P ∗(1).
(Second step: consistent estimator for h 0 .) Suppose the identifiability condition
∫
inf |σ 2 (x; h) − σ 2 (x; h 0 )| 2 μ q (dx)>0 ∀ε>0
h : d H (h,h 0 )>ε R q
is satisfied. Merely by a technical reason, assume that there exists a constant
L 4 > 0 such that E[wi 4|F i−1]

3576 Y. NISHIYAMA
PROOF OF LEMMA 5.2. Notice that ˜ n (θ 0 ,h 0 ) = 0. For any (possibly random)
sequence (θ n ,h n ) converging in outer probability to (θ 0 ,h 0 ), it holds that
˜ n (θ n ,h n ) − ˜ n (θ 0 ,h 0 )
(11)
= 1 n
= 1 n
= 1 n
n∑
i=1
n∑
i=1
n∑
i=1
Ṡ(X i ; θ n )
σ 2 (X i ; h n ) [S(X i; θ 0 ) − S(X i ; θ n )]
Ṡ(X i ; θ n )
σ 2 (X i ; h n )Ṡ(X i; θ 0 )(θ 0 − θ n ) + o P ∗(‖θ n − θ 0 ‖)
Ṡ(X i ; θ 0 )
σ 2 (X i ; h 0 )Ṡ(X i; θ 0 )(θ 0 − θ n ) + o P ∗(‖θ n − θ 0 ‖)
=−I(θ 0 ,h 0 )(θ n − θ 0 ) + o P ∗(‖θ n − θ 0 ‖).
To show (11) above, do the same argument as the proof of Lemma 4.2 using (10)
instead of (6). □
PROOF OF THEOREM 5.3. The assertions follow from Theorems 2.1 and 2.3
by using also Lemmas 5.1 and 5.2, and Theorem 3.1(ii), respectively. □
PROOF OF THEOREM 5.4. To prove the first step, we will apply Corollary
3.2.3 of van der Vaart and Wellner (1996). We can write A n (θ) = T 1,n + T 2,n (θ) +
T 3,n (θ) where
T 1,n = 1 n∑
|X i − S(X i ; θ 0 )| 2 ,
n
i=1
T 2,n (θ) = 1 n∑ ( S(Xi ; θ 0 ) − S(X i ; θ) ) σ(X i ; h 0 )w i ,
n
i=1
T 3,n (θ) = 1 n∑
|S(X i ; θ 0 ) − S(X i ; θ)| 2 .
n
i=1
The term T 1,n converges in probability to a constant C 1 . On the other hand, by using
Proposition 4.5 of Nishiyama (2000a), we have that √ nT 2,n converges weakly
in C() to a tight law, thus, sup θ∈ |T 2,n (θ)| converges in outer probability to zero.
Finally, by Theorem 3.1, it holds that sup θ∈ |T 3,n (θ) − T 3 (θ)|=o P ∗(1) where
∫
T 3 (θ) = |S(x; θ 0) − S(x; θ)| 2 μ(dx).
R q
Hence, we have sup θ∈ |A n (θ) − (C 1 + T 3 (θ))|=o P ∗(1), and van der Vaart and
Wellner’s (1996) consistency theorem yields the assertion of the first step.

SEMIPARAMETRIC Z-ESTIMATION 3577
To prove the second step, we shall apply Corollary 3.2.3 of van der Vaart and
Wellner (1996) again. Let us first see that sup h∈H |B n (h) − ˜B n (h)| =o P ∗(1)
where
˜B n (h) = 1 n∑
n
∣ |X i − S(X i ; θ 0 )| 2 − σ 2 (X i ; h) ∣ 2 .
n
i=1
Now, notice that
∣ ∣∣ |X i − S(X i ; θ LS )|2 − σ 2 (X i ; h) ∣ 2 − ∣∣ |X i − S(X i ; θ 0 )| 2 − σ 2 (X i ; h) ∣ 2∣ ∣
≤ ∣∣ |X i − S(X i ; θn
LS )|2 +|X i − S(X i ; θ 0 )| 2 − 2σ 2 (X i ; h) ∣ × ∣∣ |X i − S(X i ; θn
LS )|2 −|X i − S(X i ; θ 0 )| 2∣ ∣
≤ ∣∣ |X i − S(X i ; θn
LS )|2 +|X i − S(X i ; θ 0 )| 2 − 2σ 2 (X i ; h) ∣ ×|2X i − S(X i ; θ LS
n ) − S(X i; θ 0 )|
×|S(X i ; θ LS
n ) − S(X i; θ 0 )|
≤ L(X i , X i )‖θ LS
n
− θ 0 ‖,
where L(x 0 ,x 1 ,...,x q ) = C‖x 0 |+(x 1 ,...,x q )| 4 for a constant C. Given any
ε>0 choose M>0 such that ∫ R q+1 L(x 0,x 1 ,...,x q )1 {L(x0 ,x 1 ,...,x q )>M}μ q+1 (dx 0 ,
dx 1 ,...,dx q )M} diam().
i=1
The second term of the right-hand side converges to a positive constant which
is smaller than ε · diam(). Since the choice of ε>0 is arbitrary, we have
sup h∈H |B n (h) − ˜B n (h)|=o P ∗(1).
Now, we can write ˜B n (h) = T 1,n + T 2,n (h) + T 3,n (h), where
T 1,n = 1 n
T 2,n (h) = 2 n
T 3,n (h) = 1 n
n∑
∣ |X i − S(X i ; θ 0 )| 2 − σ 2 (X i ; h 0 ) ∣ 2 ,
i=1
n∑
i=1
n∑
i=1
σ 2 (X i ,h 0 )(w 2 i − 1)( σ 2 (X i ,h 0 ) − σ 2 (X i ; h) ) ,
|σ 2 (X i ; h 0 ) − σ 2 (X i ; h)| 2 .
The term T 1,n converges in probability to a constant C 1 by assumption. By
Jain–Marcus’ CLT for martingale difference arrays, it is easy to show that
sup h∈H |T 2,n (h)|=o P ∗(1) (here, we use the technical assumption that E[w 4 i |F i−1]

3578 Y. NISHIYAMA
is bounded). Finally, by Theorem 3.1, it holds that sup h∈H |T 3,n (h) − T 3 (h)| =
o P ∗(1) where
∫
T 3 (h) = |σ 2 (x; h 0 ) − σ 2 (x; h)| 2 μ q (dx).
R q
Consequently, we have sup h∈H |B n (h) − (C 1 + T 3 (h))|=o P ∗(1). Therefore, the
claim of the second step follows form van der Vaart and Wellner’s (1996) consistency
theorem. □
Acknowledgments. I thank the Associate Editor and the referees for their
careful reading and the advice to present a sharper version of the moment conditions.
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