Georgian Mathematical Journal
1(1994), No. 2, 197-212
LIMIT DISTRIBUTION OF THE INTEGRATED SQUARED
ERROR OF TRIGONOMETRIC SERIES REGRESSION
ESTIMATOR
E. NADARAYA
Abstract. Limit distribution is studied for the integrated squared
error of the projection regression estimator (2) constructed on the
basis of independent observations (1). By means of the obtained
limit theorems, a test is given for verifying the hypothesis on the
regression, and the power of this test is calculated in the case of
Pitman alternatives.
Let observations Y1 , Y2 , . . . , Yn be represented as
Yi = µ(xi ) + εi ,
i = 1, n,
(1)
where µ(x), x ∈ [−π, π], is the unknown regression function to be estimated
by observations Yi ; xi , i = 1, n, are the known numbers, and −π = x0 <
x1 < · · · < xn ≤ π, εi , i = 1, n, are independent equidistributed random
variables; Eε1 = 0, Eε21 = σ 2 , and Eε41 < ∞.
The problem of nonparametric estimation of the regression function µ(x)
for the model (1) has a recent history and has been treated only in few
papers. In particular, a kernel estimator of the Rosenblatt–Parzen type for
µ(x) was proposed for the first time in [1].
Assume that µ(x) is representable as a converging series in L2 (−π, π)
with respect to the orthonormal trigonometric system
o∞
n
.
(2π)−1/2 , π −1/2 cos ix, π −1/2 sin ix
i=1
Consider the estimator of the function µ(x) constructed by the projection
method of N.N. Chentsov [2]
N
a0n X
µnN (x) =
+
ain cos ix + bin sin ix,
2
i=1
1991 Mathematics Subject Classification. 62G07.
197
(2)
198
E. NADARAYA
where N = N (n) → ∞ for n → ∞ and
ain =
n
1X
Yj ∆j cos ixj ,
π j=1
bin =
n
1X
Yj ∆j sin ixj ,
π j=1
∆j = xj − xj−1 , j = 1, n,
i = 0, N .
The estimator (2) can be rewritten in a more compact way as
µnN =
n
X
j=1
where KN (u) =
1
2π
P
Yj ∆j KN (x − xj ),
eiru is the Dirichlet kernel.
|r|≤N
In [3], p.347, N.V. Smirnov considered estimators of the type (2) for
a specially chosen class of functions µ(x) in the case of equidistant points
xj ∈ [−π, π] and of independent and normally distributed observation errors
εi . In [4] an estimator of the type (2) is obtained, which is asymptotically
equivalent to projection estimators which are optimal in the sense of some
accuracy criterion. The asymptotics of the mean value of the integrated
squared error of the estimator (2) is considered in [5].
It is of interest to investigate the limit distribution of the integrated
squared error
Z
π
−π
[µnN (x) − µ(x)]2 dx,
which is the goal pursued in this paper. The method used to prove the
theorems below is based on the functional limit theorem for a sequence of
semimartingales [6].
Denote
Z π
‚
ƒ2
n
µnN (x) − EµnN (x) dx,
UnN =
2π(2N + 1) −π
Qir = ∆i ∆r KN (xi − xr ),
ηik =
2
=
σnN
n
εi εk Qik ,
π(2N + 1)σnN
ξ1 = 0, ξk =
k−1
X
ηik ,
n X
r−1
X
n2 σ 4
Q2 ,
π 2 (2N + 1)2 r=2 j=1 jr
k = 2, n, ξk = 0, k > n,
i=1
and assume that Fk is σ-algebra generated by random variables ε1 , ε2 ,
. . . , εk , F0 = (φ, Ω).
Lemma 1 ([7], p.179). The stochastic sequence (ξk , Fk )k≥1 is a martingale-difference.
LIMIT DISTRIBUTION OF THE SQUARED ERROR
199
Lemma 2. Let p(x) be the known positive continuously differentiable
density on [−π, π], and points xi be chosen from the relation
Rdistribution
xi
i
p(u)
du
=
n , i = 1, n.
−π
If
N ln N
n
→ 0 for n → ∞, then
EUnN = θ1 + O
’
2
(2N + 1)σnN
“
Z π
σ2
p−1 (u) du,
, θ1 =
(2π)2 −π
Z π
σ4
→ θ2 =
p−2 (u) du.
4π 3 −π
N ln N
n
(3)
(4)
Proof. From the definition of xi we easily obtain
”
 1 ‘•
1
1+O
,
∆i =
np(xi )
n
€ 
where O n1 is uniform with respect to i = 1, n.
Hence it follows that
Qir =
”
 1 ‘•
1
K
)
(x
−
x
1
+
O
.
N
r
i
n2 p(xi )p(xr )
n
(5)
Taking into account the relation
max |KN (u)| = O(N )
−π≤u≤π
(6)
and (5), we find
2
=
σnN
n
n X
1‘
X
1
σ4
2
KN
(xi −xj )
+O
.
2
2
2
2
2π (2N +1) n i=1 j=1
[p(xi )p(xj )]
n
(7)
Let F (x) be a distribution function with density p(x) and Fn (x) be an
empirical
Pn distribution function of the “sample” x1 , x2 , . . . , xn , i.e., Fn (x) =
n−1 k=1 I(−∞,x) (xk ), where IA (·) is the indicator of the set A. Then the
right side of (7) can be written as the integral
2
σnN
=
σ4
2π 2 (2N + 1)2
Z
π
−π
Z
π
−π
2
KN
(t − s)
1‘
dFn (t) dFn (s)
O
+
.
[p(t)p(s)]2
n
200
E. NADARAYA
Further we have
ŒZ Z
Z
Œ π π 2
dFn (t) dFn (s)
Œ
s)
−
(t
−
K
N
Œ
[p(t)p(s)]2
−π
−π
π
−π
≤ I1 + I2 ,
Z
π
−π
2
(t − s)
KN
Œ
dF (t) dF (s) ŒŒ
≤
[p(t)p(s)]2 Œ
Œ
ŒZ π Z π
Œ
ƒŒ
dFn (s) ‚
2
Œ
I1 = Œ
KN (t − s)
dFn (t) − dF (t) ŒŒ ,
[p(t)p(s)]2
−π −π
ŒZ Z
Œ
Œ π π 2
ƒŒ
dF (t) ‚
Œ
I2 = Œ
KN (t − s)
dFn (s) − dF (s) ŒŒ .
[p(t)p(s)]2
−π −π
By integration by parts in the internal integral in I1 we readily obtain
Z
Z π
Œ Œ€ 0
dFn (s) 𠌌
(t − s)p(t) −
dFn (t) − dF (t)Œ Œ KN
I1 ≤ 2
2
−π
−π p (s)
Œ

(8)
−KN (t − s)p0 (t) KN (t − s)/p3 (t)Œ dt.
€1
Since sup−π≤x≤π |Fn (x) − F (x)| = O n and the relations [8]1
Z π
2
0
KN
(u) du = 2N + 1,
(u)| = O(N 2 ),
max |KN
−π≤u≤π
−π
(9)
Z π
|KN (u)| du = O(ln N )
−π
are fulfilled, from (8) we have the estimate
’ 2
“
N ln N
I1 = O
.
n
In the same manner we show that
I2 = O
Therefore
2
(2N +1)σnN
σ4
= 3
4π
Z
π
−π
Z
π
’
N 2 ln N
n
“
.
dt ds
ΦN (s−t)
+O
p(s)p(t)
−π
’
N ln N
n
“
,
(10)
2
where ΦN (u) = 2N2π+1 KN
(u) is the Fejér kernel.
We shall complete the definition of the function p−1 outside [−π, π] as
regards its periodicity and also note that KN (u) and ΦN (u) are periodic
functions with the period 2π. The continued function will be denoted by
g(x). Then
Z π
Z π Z π
dt ds
ΦN (s − t)
=
p−2 (x) dx + χn ,
p(s)p(t)
−π −π
−π
1 See
p. 115 in the Russian version of [8]: “Mir”, Moscow, 1965.
LIMIT DISTRIBUTION OF THE SQUARED ERROR
201
where
|χn | ≤
Z
π
|σ̄N (x) − g(x)|dx,
−π
π
σ̄N (x) =
Z
−π
ΦN (u)g(x − u)du.
Hence, on account of the theorem on convergence of the Fejér integral
σ̄N (x) to g(x) in the norm of the space L1 (−π, π) (see [9], p.481), we have
χn → 0 for n → ∞.
Therefore
Z π
σ4
2
p−2 (x) dx.
(2N + 1)σnN →
4π 3 −π
Now we shall prove (3). We have
DµnN (x) = σ
2
n
X
j=1
”
 1 ‘•
1
2
K (x − xj ) 1 + O
.
np2 (xj ) N
n
Applying the same reasoning as in deriving (10), we find
Z
 N 2 ln N ‘
σ2 π 2
ds
KN (x − s)
DµnN (x) =
+O
.
n −π
p(s)
n2
(11)
Therefore
Z
Z
 N ln N ‘
ds dt
+O
=
p(s)
n
−π −π
Z π
 N ln N ‘
σ2
=
.
p−1 (s) ds + O
2
(2π) −π
n
EUnN =
σ2
(2π)2
π
π
ΦN (t − s)
d
Denote by the symbol → the convergence in distribution, and let ξ be a
random variable having normal distribution with zero mean and variance 1.
2
Theorem 1. Let xi i = 1, n be the same as in Lemma 2 and N nln N → 0
√
−1/2 d
→ ξ.
for n → ∞. Then, as n increases, 2N + 1(UnN − θ1 )θ2
Proof. We have
UnN − EUnN
= Hn(1) + Hn(2) ,
σnN
where
Hn(1) =
n
X
j=1
ξj ,
Hn(2) =
n
X
n
(ε2 − Eε2i )Qii .
2π(2N + 1)σnN i=1 i
202
E. NADARAYA
(2)
Hn converges to zero in probability. Indeed,
n
X
n2 Eε14
2
Qii
=
2
(2π)2 (2N + 1)2 σnN
i=1
’
n
 1 ‘“
X
Eε41
1
2
=
≤
K (0) 1 + O
2 · n2
(2π)2 (2N + 1)2 σnN
(p(xi ))4 N
n
i=1
N ‘
1
≤C 2 =O
,
nσnN
n
DHn(2) ≤
(2)
P
whence Hn → 0. Here and in what follows C is the positive constant
varying from one formula to another and the letter P above the arrow
denotes convergence in probability.
(1) d
We will now prove that Hn → ξ. To this end we will verify the validity
of Corollaries 2 and 6 of Theorem 2 from [6]. We have to show whether
the conditions contained in these statements are fulfilled for asymptotic
normality of the square-integrable martingale-difference, which, by Lemma
1, is our sequence {ξk , Fk }k≥1 .
Pn
A direct calculation shows that k=1 Eξk2 = 1. Asymptotic normality
will take place if for n → ∞
n
X
‚
ƒ
E ξk2 · I(|ξk | ≥ ε) | Fk−1 → 0
(12)
k=1
and
n
X
k=1
P
ξk2 → 1.
(13)
P
It is shown in [6] that the fulfillment of (13) and the condition sup |ξk | → 0
1≤k≤n
implies the validy of (12) as well.
Since for ε > 0
P
(1)
d
ˆ
n
X
‰
Eξk4 ,
sup |ξk | ≥ ε ≤ ε−4
1≤k≤n
k=1
to prove Hn → ξ we have to verify only (13) by the relation (15) to be
given below.
Pn
P
We will establish k=1 ξk2 → 1. For this itPsuffices to make sure that
P
n
n
2
2
E( k=1 ξk − 1)) → 0 for n → ∞, i.e., due to i=1 Eξi2 = 1
E
n
X
k=1
ξk2
‘2
=
n
X
k=1
Eξk4 + 2
X
1≤k1 <k2 ≤n
Eξk21 ξk22 → 1.
(14)
LIMIT DISTRIBUTION OF THE SQUARED ERROR
203
Pn
In the first place we find that k=1 Eξk4 → 0 for n → ∞. By virtue of
the definitions of ξk and ηij we write
n
X
Eξk4 = Ln(1) + L(2)
n ,
k=1
where
n k−1
L(1)
n =
XX
n4
Q4jk ,
Eε41 (Eε41 − 3σ 4 )
4
4
4
π (2N + 1) σnN
j=1
k=2
Ln(2) =
3n4 σ 4 Eε41
4 π4
(2N + 1)4 σnN
n  k−1
X
X
k=2
j=1
Q2jk
‘2
.
From (5) and (6) we obtain
|L(1)
n |=C
and also
|L(2)
n |=C
1
4
n2 N 4 σnN
≤C
=C
n
X
k=2
1
4
n2 N 4 σnN
1
4
n2 N 4 σnN
≤C
+
n k−1
 1 ‘i
X K 4 (xj − xk ) h
X
N
1+O
≤
4
4
4
4
n N σnN
[p(xj )p(xk )]
n
k=2 j=1
 N ‘2 ‘
−4
≤ Cn−2 σnN
=O
n
1
hZ
1
−π
n
X
k=1

1
n
n ’Z π
X
−π
k=2
n šh
X
4
n2 N 4 σnN
π
2
k−1
 1 ‘i
X K 2 (xj − xk ) h
1
N
 ≤

1+O
n j=1 p(xj )p(xk )
n

k=2
k−1
X
j=1
2
2
(xj − xk ) =
KN
“2
2
KN
(xk − u)
(u)
≤
dF
n
p2 (u)
i2
2
KN
(xk − u)p−1 (u) du +
›
i2 2
€
2
.
KN
(xk − u)p−2 (u) d Fn (u) − F (u)
Hence, taking into accountthe
‘ relation (9) and the formula of integration
(2)
by parts, we have |Ln | = O n1 . Therefore
n
X
k=1
Eξk4 → 0 for n → ∞.
(15)
204
E. NADARAYA
P
Let us now establish that 2
definition of ξi implies
ξk21 ξk22
=
1 −1
 kX
2
ηik
1
i=1
+
2 −1
 kX
2
ηik
2
i=1
2 −1
‘ kX
1≤k1 <k2 ≤n
2
ηik
2
i=1
1 −1
‘ kX
ηsk1 ηtk1 +
s6=t=1
=
‘
(1)
Bk1 k2
+
‘
+
Eξk21 ξk22 → 1 for n → ∞. The
1 −1
 kX
2
ηik
1
i=1
1 −1
 kX
i6=s=1
ηsk1 ηtk1
s6=t=1
(2)
Bk1 k2
+
2 −1
‘ kX
‘
ηik2 ηsk2 +
2 −1
‘ kX
k6=r=1
(3)
Bk 1 k 2
+
(4)
Bk1 k2 .
‘
ηkk1 ηrk2 =
Therefore
2
X
Eξk21 ξk22 =
A(i)
n =2
An(i) ,
i=1
1≤k1 <k2 ≤n
where
4
X
X
(i)
EBk1 k2 ,
i = 1, 4.
1≤k1 <k2 ≤n
(3)
In the first place we consider An . By the definition of ηij we obtain
2
6 t, k1 < k2 . Thus
η η = 0, s =
Eηik
2 sk1 tk1
An(3) = 0.
(16)
(2)
(2)
Let us derive an estimate of An . Divide the sum EBk1 k2 into two parts:
(2)
EBk1 k2 =
kX
1 −1
k1
X
2
Eηik
η η +
1 rk2 sk2
i=1 r6=s=1
kX
1 −1
kX
2 −1
2
Eηik
η η .
1 rk2 sk2
i=1 r6=s=k1 +1
The second term is equal to zero, since i cannot coincide with r or with s
2
2
and r =
η η
= 0, and Eηik
6 s; in this case Eηik
η η
= 0 also in the
1 rk2 sk2
1 rk2 sk2
first term each time except for the case s = k1 or r = k1 .
Thus
kX
1 −1
€ 2

(2)
EBk1 k2 = 2
E ηik
η η
.
1 ik2 k1 k2
i=1
Hence, using the definition of ηij and the inequality |Qij | ≤ C nN2 obtained
from (5) and (6), we find
Œ
Œ
ŒEB (2) Œ ≤ C
k1 k2
kX
1 −1
1
Q2ik1 .
4
(2N + 1)2 σnN
i=1
(17)
LIMIT DISTRIBUTION OF THE SQUARED ERROR
205
Next, taking into account statement (4) of Lemma 2 and the definition
2
of σnN
, from (17) we have
|A(2)
n |≤C
n kX
1 −1
N ‘
X
n
1
2
Q
≤
=
O
.
C
ik1
4
2
N 2 σnN
nσnN
n
i=1
(18)
k1 =2
(4)
Consider now An . By the definition of ηij we obtain
X
8n4
Qsk1 Qsk2 Qtk1 Qtk2 ≤
4
4
4
π (2N + 1) σnN
s<t<k1 <k2
Œ
n4 hŒŒ X
Œ
≤C 4 4 Œ
Qsk1 Qsk2 Qtk1 Qtk2 Π+
N σnN
s,t,k1 ,k2
Œi
ΠX
Œ Œ
Œ
Œ
ΠΠX
+Œ
Qk1 t Qst Qk1 s Qss Π=
Q2k1 s Qk21 t Œ + Œ
A(4)
n =
k1 ,s,t
k1 ,s,t
ƒ
n4 ‚
= C 4 4 |E1 | + |E2 | + |E3 | .
N σnN
(19)
According to (5) and (6) we write
E1 = n−7
×
Z
X
s,t,k1
π
−π
KN (xs − xk1 )KN (xt − xk1 ) ×
KN (xs − u)KN (xt − u) dFn (u) + O
N2 ‘
n
.
Hence, integrating by parts and taking into account (9), we obtain
−7
E1 = n
Z
π
X
−π s,t,k
1
KN (xs − xk1 )KN (xt − xk1 ) ×
KN (xs − u)KN (xt − u)p(u) du + O
 N 4 ln N ‘
.
n5
(20)
Applying the same operations three times, we represent (20) in the form
E1 = n−4
Z
π
−π
Z
π
−π
Z
π
−π
Z
π
−π
KN (z − u)KN (z − t)KN (y − u)KN (y − t) ×
×p(y)p(u)p(z)p(t) du dt dz dy + O
 N 4 ln N ‘
n5
 N ln3 N ‘
 N 4 ln N ‘
+O
.
=O
n4
n5
=
206
E. NADARAYA
Thus
 ln3 N ‘
 N 2 ln N ‘
n4
=
O
|E
|
+
O
.
1
4
N 4 σnN
N
n
(21)
N2 ‘
n4
|E
,
|
=
O
2
4
N 4 σnN
n
N2 ‘
n4
|
|E
.
=
O
3
4
N 4 σnN
n
(22)
Further, it is not difficult to show
Therefore (19), (21), and (22) imply
A(4)
n =O
 N 2 ln N ‘
n
+O
(1)
 ln3 N ‘
N
.
(23)
(1)
Finally, we will show that An → 1 for n → ∞. For this represent An
(1)
(1)
(2)
in the form An = Qn + Qn , where
Q(1)
n
=2
1 −1
X  kX
i=1
k1 <k2
Qn(2) = 2
 X
k1 <k2
2
Eηik
1
2 −1
‘ kX
j=1
(1)
EBk1 k2 −
‘
2
Eηik
,
2
1 −1
X  kX
k1 <k2
2
Eηik
1
i=1
2 −1
‘ kX
2
Eηik
2
j=1
‘‘
.
2
it follows that
From the definition of σnN
Q(1)
n =1−
where
n  k−1
X
X
k=2
Therefore
k=2
2
Eηik
i=1
‘2
,
n4 X  X 2 ‘2
Qik ≤
4
N 4 σnN
k=2 i=1
N2 ‘
1
≤C 4 =O
.
nσnN
n
2
Eηik
i=1
‘2
n  k−1
X
X
n
k−1
≤C
€ 2 
Q(1)
n = 1 + O N /n .
(2)
(2)
Let us now show that Qn → 0. Qn can be written as
Q(2)
n =2
1 −1
X h kX
€
k1 <k2
i=1
i
2
2
2
2
cov(ηik
.
,
η
)
+
cov(η
,
η
)
ik2
ik1 k1 k2
1
(24)
LIMIT DISTRIBUTION OF THE SQUARED ERROR
207
But
n4
2
Eηik
η2 ≤ C
1 ik2
≤C
€
1
4
n4 N 4 σnN
4
N 4 σnN

4
max |KN (u)| = O
−π≤u≤π

€
−2
2
. Therefore
Similarly, Eηij
= O n−2 σnN
2
2
cov(ηik
)=O
, ηik
1
2
Further, since
Q2ik1 · Q2ik2 ≤
P
1≤k1 <k2 ≤n (k1

1 ‘
.
4
n4 σnN
1 ‘
.
4
n4 σnN
(25)
− 1) = O(n3 ), (25) implies
Qn(2) = O
Thus, according to (24) and (26)
N2 ‘
n
(26)
.
€ 2 
A(1)
n = 1 + O N /n .
(27)
Combining the relations (16), (18), (23) and (27), we finally obtain
E
n
X
k=1
‘2
ξk2 − 1
→ 0 for n → ∞.
Therefore
UnN − EUnN d
→ ξ.
σnN
N
2
→ θ2 ,
) and (2N + 1)σnN
Further, due to Lemma 2, EUnN = θ1 + O( N ln
n
and hence we obtain
−1/2 d
(2N + 1)1/2 (UnN − θ1 )θ2
Denote
TnN
n
=
2π(2N + 1)
Z
π
−π
→ ξ.
ƒ2
‚
µnN (x) − µ(x) dx.
Theorem 2. Let xi , i = 1, n, be the same as in Lemma 2 and the function µ(x) with period 2π have bounded derivatives up to the second order.
Moreover, if N 2 ln N/n → 0 and n ln2 N/N 9/2 → 0 for n → ∞, then
√
−1/2 d
→ ξ.
2N + 1(TnN − θ1 )θ2
208
E. NADARAYA
Before we proceed to proving the theorem, we have to show
Z π
0
(u)|du = O(N ln N ).
|KN
(28)
−π
e ν (u) =
Denote D
we have
Pν
k=1
sin ku. Then by virtue of the Abel transformation
0
KN
(u) = −
N
X
k sin ku =
N
−1
X
ν=1
k=1
e ν (u) + N D
eN .
D
Rπ
e ν (u)|du → 1 for ν → ∞.
It is well known [8] that ην = (ln ν)−1 −π |D
PN −1
Denote bN = ν=1 ln ν. Then by the Toeplitz lemma
RN =
N −1
1 X
ln ν · ην → 1.
bN ν=1
Therefore
Z
π
−π
0
|KN
(u)|du ≤
N
−1 Z π
X
ν=1
= bN · RN + N
−π
Z
π
e ν (u)|du + N
|D
−π
Z
π
−π
e N (u)|du =
|D
e N (u)|du = O(N ln N ).
|D
Let us return to the proof of the theorem. We have
A1n
TnN = UnN + A1n + A2n ,
π ‚
ƒ
ƒ‚
n
µnN (x) − EµnN (x) EµnN (x) − µ(x) dx,
=
π(2N + 1) −π
Z π
‚
ƒ2
n
A2n =
EµnN (x) − µ(x) dx.
2π(2N + 1) −π
Z
It is not difficult to find
n
hZ π
X
nσ 2
√
∆2j
Kn (y − xj ) ×
2π 2N + 1 j=1
−π
€
 i2 ‘1/2
× EµnN (y) − µ(x) dy
.
√
2N + 1E|A1n | ≤
But
EµnN (y) =
Z
π
−π
µ(x)

€ 1 ‘
1
KN (y − x) dFn (x) 1 + O
p(x)
n
LIMIT DISTRIBUTION OF THE SQUARED ERROR
and
Z
209
π
µ(x)p−1 (x)KN (y − x) dFn (x) =
Z π
1 Z π
‘
0
|KN
(u)|du .
=
µ(x)KN (y − x) + O
n −π
−π
−π
It is well known ([10], p.22) that
Z π
 ln N ‘
µ(x)KN (y − x) dx = µ(y) + O
N2
−π
uniformly in y ∈ [−π, π]. By virtue of (28) this gives us
 N ln N ‘
 ln N ‘
+O
.
EµnN (x) = µ(x) + O
2
N
n
Therefore
h n ln2 N ‘1/2 ln N
√
2N + 1E|A1n | ≤ C
+
N 9/2
N 1/4
 N 2 ln N ‘1/2 ln3/2 N i
√
+
→ 0.
n
N
(29)
(30)
Further, from (29) we have
 n ln2 N
√
N 2 ln2 N ‘
√
2N + 1A2n ≤ C
+
→ 0.
n
N 9/2
N
(31)
Finally, the statement of Theorem 2 directly follows from Theorem 1, (30),
and (31).
Using Theorems 1 and 2, it is easy to solve the problem concerning testing
of the hypothesis on µ(x). Given σ 2 , it is required to verify the hypothesis
H0 : µ(x) = µ0 (x). The critical region is defined approximately by the
inequality UnN ≥ dn (α) or TnN ≥ dn (α), where

€
dn (α) = σ 2 L1 + (2N + 1)−1/2 L2 λα ,
Z π
 1 Z π
‘1/2
−2
L1 = ((2π)−2
p−1 (x) dx, L2 =
dx
p
(x)
,
4π 3 −π
−π
normal distribution.
and λα is the quantile of level α of standard
√
Let now σ 2 be unknown. We call an N -consistent estimate of variance
σ 2 , for instance,
n
2
1 X€
Sn2 =
Yi − µnλ (xi ) ,
n i=1
where λ = λ(n) → ∞ is a sequence such that
N λ4
n → 0 for n → ∞.
λ
N
→ 0,
N ln2 λ
λ4
→ 0 and
210
E. NADARAYA
Indeed, using the expressions (11) and (29), we easily find
Denote
 N λ ‘1/2 ‘
 N 1/2 ln λ ‘
√
N (ESn2 − σ 2 ) = O
+O
.
n
λ2
(32)
Zj = Yj − Rj ,
Rj =
n
X
k=1
Then
n2 DSn2 =
n
X
j=1
Yk ∆k Kλ (xj − xk ).
DZj2 +
X
cov(Zi2 , Zi21 ).
i6=i1
Simple calculations show that cov(Zj2 , Zj21 ) = O
 4‘
√
P
O λn . This and (32) imply N (Sn2 − σ 2 ) → 0.
λ
n

λ4
n
‘
. Therefore DSn2 =
Corollary. Let the conditions of Theorem 2 be fulfilled. Moreover, let
4
2
→ 0, Nnλ → 0 and N λln4 λ → 0. Then
√
d
Sn−2 L−1
2N + 1(UnN − Sn2 L1 ) → ξ,
2
√
d
2N + 1(TnN − Sn2 L1 ) → ξ.
Sn−2 L−1
2
This corollary enables one to construct a test for verifying H0 : µ(x) =
µ0 (x). The critical region is defined approximately by the inequality UnN ≥
den (α) or TnN ≥ den (α), where den (α) is obtained from dn (α) by using Sn2
instead of σ 2 .
Consider now the local behavior of the test power in the case where the
critical region is of the form {x ∈ R1 , x ≥ dn (α)}. More exactly, find a
distribution of the quadratic functional UnN under a sequence of alternatives
close to the hypothesis H0 : µ(x) = µ0 (x). The sequence is written as
H1 : µ̄(x) = µ0 (x) + γn ϕ(x) + o(γn ),
(33)
where γn → 0 appropriately and o(γn ) is uniform in x ∈ [−π, π].
Theorem 3. Let µ̄n (x) satisfy the conditions of Theorem 2. If 2N +
1 = nδ , γn = n−1/2+δ/4 , 92 < δ < 12 , then under the alternative H1
1/2
the
(2N √+ 1)
€ 1 Rstatistic
 (UnN − θ1 ) is distributed in the limit normally
π
2
θ
ϕ
(u)du,
.
2
2π −π
LIMIT DISTRIBUTION OF THE SQUARED ERROR
211
Proof. Let us represent UnN as the sum
Z π
€
2
n
µnN (x) − E1 µnN (x) dx +
UnN =
2π(2N + 1) −π
Z π
‚
ƒ
n
µnN (x) − E1 µnN (x) ϕ
en (x) dx +
+
γn
π(2N + 1)
−π
Z π
n
+
γ2
ϕ
e2 (x) dx = A1 (n) + A2 (n) + A3 (n),
2π(2N + 1) n −π n
where E1 (·) denotes the mathematical expectation under the hypothesis H1 ,
ϕ
en (x) =
n
X
j=1
ϕ(xj )∆j Kn (x − xj ).
√
Due to Theorem 1 one can readily assertain
that 2N + 1(A1 (n) − θ1 ) is
√
distributed asymptotically normal (0, θ2 ).
By analogy with the proof of Lemma 2 we find
Z π Z π
‘2
 N 2 ln N ‘
√
1
2N + 1A3 (n) =
ϕ(y)KN (x − y)dy dx + O
.
2π −π
n
−π
Hence, by virtue of theorem 2 from [9], p.474, we have
Z π
√
1
2N + 1A3 (n) →
ϕ2 (u)du.
2π −π
Further, for our choice of N and γn we can show by simple calculations that
 ln2 n
√
ln n ‘
2N + 1E|A2 (n)| ≤ C δ/4 + 1−7δ/4 .
n
n
Thus the local behaviour of the power PH1 (UnN ≥ dn (α)) is
Z π
‘

€

−1/2 1
ϕ2 (u) du .
(34)
PH1 UnN ≥ dn (α) → 1 − Φ λα − θ2
2π −π
Rπ
Since −π ϕ2 (u)du > 0 and is equal to zero iff ϕ(x) = 0, from (34)
we conclude that the test for the hypothesis H0 : µ(x) = µ0 (x) against
alternatives of the form (33) is asymptotically strictly unbiased.
Remark. Similar results can be obtained by the same method for the
kernel estimator of Priestley and Chao [1].
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Roy. Statist. Asoc. ser. B 34(1972), 385-392.
212
E. NADARAYA
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(Received 1.07.1992)
Author’s address:
Faculty of Mechanics and Mathamatics
I.Javakhishvili Tbilisi State University
2 University St., 380043 Tbilisi
Republic of Georgia