Про міру інтегрального квадратичного відхилення із узагальненою вагою для оцінки щільності розподілу ймовірностей Розеньлатта - Парзена
Про міру інтегрального квадратичного відхилення із узагальненою вагою для оцінки щільності розподілу ймовірностей Розеньлатта - Парзена
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Інститут математики НАН України
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irk-123456789-1659602020-02-18T01:27:06Z Про міру інтегрального квадратичного відхилення із узагальненою вагою для оцінки щільності розподілу ймовірностей Розеньлатта - Парзена Бабілуа, П.К. Надарая, Е.А. Сохадзе, Г.А. Статті Про міру інтегрального квадратичного відхилення із узагальненою вагою для оцінки щільності розподілу ймовірностей Розеньлатта - Парзена On a measure of integral square deviation with generalized weight for the Rosenblatt–Parzen probability density estimator 2010 Article Про міру інтегрального квадратичного відхилення із узагальненою вагою для оцінки щільності розподілу ймовірностей Розеньлатта - Парзена / П.К. Бабілуа // Український математичний журнал. — 2010. — Т. 62, № 4. — С. 514–535. — Бібліогр.: 6 назв. — укр. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165960 519.21 uk Український математичний журнал Інститут математики НАН України |
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Статті Статті |
| spellingShingle |
Статті Статті Бабілуа, П.К. Надарая, Е.А. Сохадзе, Г.А. Про міру інтегрального квадратичного відхилення із узагальненою вагою для оцінки щільності розподілу ймовірностей Розеньлатта - Парзена Український математичний журнал |
| description |
Про міру інтегрального квадратичного відхилення із узагальненою вагою для оцінки щільності розподілу ймовірностей Розеньлатта - Парзена |
| format |
Article |
| author |
Бабілуа, П.К. Надарая, Е.А. Сохадзе, Г.А. |
| author_facet |
Бабілуа, П.К. Надарая, Е.А. Сохадзе, Г.А. |
| author_sort |
Бабілуа, П.К. |
| title |
Про міру інтегрального квадратичного відхилення із узагальненою вагою для оцінки щільності розподілу ймовірностей Розеньлатта - Парзена |
| title_short |
Про міру інтегрального квадратичного відхилення із узагальненою вагою для оцінки щільності розподілу ймовірностей Розеньлатта - Парзена |
| title_full |
Про міру інтегрального квадратичного відхилення із узагальненою вагою для оцінки щільності розподілу ймовірностей Розеньлатта - Парзена |
| title_fullStr |
Про міру інтегрального квадратичного відхилення із узагальненою вагою для оцінки щільності розподілу ймовірностей Розеньлатта - Парзена |
| title_full_unstemmed |
Про міру інтегрального квадратичного відхилення із узагальненою вагою для оцінки щільності розподілу ймовірностей Розеньлатта - Парзена |
| title_sort |
про міру інтегрального квадратичного відхилення із узагальненою вагою для оцінки щільності розподілу ймовірностей розеньлатта - парзена |
| publisher |
Інститут математики НАН України |
| publishDate |
2010 |
| topic_facet |
Статті |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/165960 |
| citation_txt |
Про міру інтегрального квадратичного відхилення із узагальненою вагою для оцінки щільності розподілу ймовірностей Розеньлатта - Парзена / П.К. Бабілуа // Український математичний журнал. — 2010. — Т. 62, № 4. — С. 514–535. — Бібліогр.: 6 назв. — укр. |
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Український математичний журнал |
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UDC 519.21
E. Nadaraya, P. Babilua (Tbilisi State Univ., Georgia),
G. Sokhadze (A. Tsereteli Kutaisi State Univ., Georgia)
ON AN INTEGRAL SQUARE DEVIATION MEASURE
WITH THE GENERALIZED WEIGHT
OF THE ROSENBLATT – PARZEN PROBABILITY
DENSITY ESTIMATOR
ПРО МIРУ IНТЕГРАЛЬНОГО КВАДРАТИЧНОГО
ВIДХИЛЕННЯ IЗ УЗАГАЛЬНЕНОЮ ВАГОЮ
ДЛЯ ОЦIНКИ ЩIЛЬНОСТI РОЗПОДIЛУ ЙМОВIРНОСТЕЙ
РОЗЕНБЛАТТА – ПАРЗЕНА
The limit distribution of an integral square deviation with the weight of “delta-functions” of the Rosenblatt –
Parzen probability density estimator is defined. Also, the limit power of the goodness-of-fit test constructed
by means of this deviation is investigated.
Встановлено граничний розподiл iнтегрального квадратичного вiдхилення з вагою типу дельта-функцiй
для оцiнки щiльностi розподiлу ймовiрностей Розенблатта – Парзена. Також дослiджено граничну по-
тужнiсть критерiю, побудованого за допомогою цього вiдхилення.
It is well known that the limit distributions of some global measures of deviation of esti-
mates fn(x) of a density f(x), for example, the integral quadratic deviation constructed
by means of the so-called weight function W (x) not depending on n were studied in the
works of P. Bickel and M. Rosenblatt [1], M. Rosenblatt [2], E. Nadaraya [3], P. Hall
[4] and others.
In T. Tony Cai and Mark G. Low [5], the theory of obtaining the asymptotic behavior
of the mean square error
R(fn, f ;Wn) = E
∫
Wn(x)
(
fn(x)− f(x)
)2
dx, (1)
is developed, whereWn(x) = anW (an(x−`0)), {an} is a sequence of positive integers,
W (x) ≥ 0 is a Borel-measurable function and `0 is some fixed point.
If in (1) we take W (x) =
1
2
I (−1 ≤ x ≤ 1), and pass to the limit as an → ∞ for
fixed n, then, roughly speaking,
R(fn, f ;Wn) ' E
(
fn(`0)− f(`0)
)2
,
i.e., we come to the mean square error of the nonparametric estimate of the density
fn(x) at the point `0.
If however in (1) we take an ≡ 1 for all n, `0 = 0 and assume that W (x) ≥ 0 is an
arbitrary bounded function, then
R(fn, f ;W ) = E‖fn − f‖2L2(W ),
i.e., we obtain a usual integral mean square error of the estimate fn(x). Therefore
the value R(fn, f ;Wn) can be regarded as a generalization of the measure of density
estimation precision covering a mean square deviation of the estimate of the density
c© E. NADARAYA, P. BABILUA, G. SOKHADZE, 2010
514 ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 4
ON AN INTEGRAL SQUARE DEVIATION MEASURE WITH THE GENERALIZED WEIGHT . . . 515
fn(x) at the fixed point and an integral mean square deviation. Hence it is natural to
pose the question on the limit distribution of the value ‖fn − f‖2L2(Wn)
, Wn(x) =
= anW (an(x− `0)).
Let us give the corresponding result for the case where fn(x) is the nonparametric
estimate of the Rosenblatt – Parsen density distribution and an → ∞ as n → ∞. The
case an → a0 <∞ is of no interest since it follows from the results of [1 – 4].
Let X1, X2, . . . , Xn be independent, equally distributed random values having the
unknown density function of f(x). Assume that the sought density f(x) ∈ L2(Wn)
(Wn(x) is a weight function) and consider the ways of empirical approximation of this
density when measuring the error value in the metric L2(Wn) of the following form:
fn(x) =
λn
n
n∑
i=1
K
(
λn(x−Xi)
)
,
where K(x) is a function belonging to the class of functions
H =
{
K : K(x) ≥ 0,
∫
K(x) = 1, K(−x) = K(x),
sup
x∈(−∞,∞)
K(x) <∞, x2K(x) ∈ L1(−∞,∞)
}
,
and {λn} is a sequence of numbers converging to infinity.
Denote by F the set of bounded functions on (−∞,∞) having bounded derivatives
up to second order inclusive.
In this paper we consider the problem of finding the limit distribution of the functional
Un =
n
λn
∫ (
fn(x)− f(x)
)2
Wn(x) dx.
We also study the properties of the power of the goodness-of-fit test constructed by
means of the statistic Un.
1. Limit distribution of Un. We will need the following notation:
U (1)
n = n
∫ (
fn(x)− Efn(x)
)2
Wn(x) dx, ∆n(f) = EU (1)
n ,
αn(x, y) = λn
[
K
(
λn(x− y)
)
− EK
(
λn(x−X1)
)]
,
σ2
n(f) = 2
∫∫ (
Eαn(u1, X1)αn(u2, X1)
)2
Wn(u1)Wn(u2) du1 du2,
Wn(x) = anW
(
an(x− `0)
)
,
η
(n)
ij =
2
nσn(f)
∫
αn(x,Xi)αn(x,Xj)Wn(x) dx,
ξ
(n)
j =
j−1∑
i=1
η
(n)
ij , j = 2, . . . , n,
ξ
(n)
1 = 0, ξ
(n)
j = 0, j > n,
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 4
516 E. NADARAYA, P. BABILUA, G. SOKHADZE
Y
(n)
k =
k∑
i=1
ξ
(n)
i , Fk = σ(ω : X1, . . . , Xk),
where Fk is the σ-algebra generated by random values X1, X2, . . . , Xk and F0 =
= {∅,Ω}.
In the sequel, for the sake of simplicity, instead of ξ(n)j , η
(n)
ij and Y (n)
j we will write
respectively ξj , ηij and Yj .
Lemma 1. A stochastic sequence (Yj ,Fj)j≥1 is a martingale, while a sequence
(ξj ,Fj)j≥1 is a difference-martingale.
The proof follows from the representation
E
(
Yj+1 | Fj
)
= E
(
j+1∑
i=1
ξi | Fj
)
=
= E
(
j∑
i=1
ξi | Fj
)
+ E
(
ξj+1 | Fj
)
= Yj a.s.,
since E(ξj+1 | Fj) = 0 and for all j ≥ 1 we have E|Yj | <∞.
Furthermore, since ξj+1 = Yj+1 − Yj and E(ξj+1 | Fj) = 0 a.s., (ξj ,Fj)j≥1 is a
difference-martingale.
Lemma 2. Let K(x) ∈ H, f(x) ∈ F, W (x) be bounded and W (x) ∈ L2(R). If
λn →∞, an →∞ and an/λn → 0 as n→∞, then
(λnan)−1σ2
n(f) −→ σ2(f) = 2f2(`0)
∫
K2
0 (z) dz
∫
W 2(v) dv,
where K0 = K ∗K, f(`0) 6= 0.
Proof. We have
σ2
n(f) = 2λ4n
∫∫ [
λ−1n
∫
K(t)K
(
λn(u2 − u1)− t
)
f
(
u1 −
t
λn
)
dt−
−λ−2n
∫
K(t1)f
(
u1 −
t1
λ
)
t1
∫
K(t2)f
(
u2 −
t2
λn
)
dt2
]2
×
×a2nW
(
an(u1 − `0)
)
W
(
an(u2 − `0)
)
du1 du2. (2)
Next performing the change of variables in (2) we obtain
σ2
n(f) = In1 + In2 + In3,
where
In1 = 2λna
2
n
∫∫ [ ∫
K(t)K(z − t)f
(
u1 −
t
λn
)
dt
]2
×
×W
(
an(u1 − `0)
)
W
(
an
(
u1 +
z
λn
− `0
))
du1 dz,
In2 = −4a2n
∫∫ [ ∫
K(t)K(z − t)f
(
u1 −
t
λn
)
dt×
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 4
ON AN INTEGRAL SQUARE DEVIATION MEASURE WITH THE GENERALIZED WEIGHT . . . 517
×
∫
K(t1)f
(
u1 −
t1
λn
)
dt1
∫
K(t2)f
(
u1 +
z
λn
− t2
λn
)
dt2
]
×
×W
(
an(u1 − `0)
)
W
(
an
(
u1 +
z
λn
− `0
))
du1 dz,
In3 = 2λ−1n a2n
∫∫ [ ∫
K(t1)f
(
u1 −
t1
λn
)
dt1×
×
∫
K(t2)f
(
u1 +
z
λn
− t2
λn
)
dt2
]2
×
×W (an(u1 − `0))W
(
an
(
u1 +
z
λn
− `0
))
du1 dz.
It is easy to see that
In2 ≤ 4c1a
2
n
∫ [ ∫
K(t)K(z − t) dt×
×
(∫
K(t1) dt1
)2 ∫
W (an(u1 − `0)) du1
]
dz ≤
≤ c2an,
In3 ≤ 2c3λ
−1
n a2n
∫ [
W (an(u1 − `0))×
×
∫
W
(
an
(
u1 −
z
λn
− `0
))
dz
]
du1 ≤ c4.
Therefore,
σ2
n(f) = In1 +O(an) +O(1),
and also
In1 = 2λna
2
n
∫∫
f2(u1)K2
0 (z)W (an(u1 − `0))×
×W
(
an
(
u1 +
z
λn
− `0
))
du1 dz +An1 +An2,
An1 = 2λna
2
n
∫∫ [ ∫
K(t)K(z − t)
(
f
(
u1 −
t
λn
)
− f(u1)
)
dt
]2
×
×W (an(u1 − `0))W
(
an
(
u1 +
z
λn
− `0
))
du1 dz ≤
≤ c5
an
λn
∫
t2K(t) dt = O
(
an
λn
)
,
An2 ≤ c6λna2n
∫∫ [ ∫
K(t1)K(z − t1)
|t1|
λn
dt1
∫
K(t2)K(z − t2) dt2
]
×
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 4
518 E. NADARAYA, P. BABILUA, G. SOKHADZE
×W (an(u1 − `0))W
(
an
(
u1 +
z
λn
− `0
))
du1 dz ≤
≤ c7a2n
∫ [ ∫
|t|K(t) dt
∫
W (an(u1 − `0)) du1
]
≤ c8an.
Thus
(λnan)−1σ2
n(f) = 2
∫∫
f2
(
`0 +
v
an
)
K2
0 (z)W 2(v) dv dz+
+An3 +O
(
1
λn
)
+O
(
1
λnan
)
, (3)
where
An3 = 2
∫∫
f2
(
`0 +
v
an
)
K2
0 (z)W 2(v)
[
W
(
v +
an
λn
z
)
−W (v)
]
dv dz,
and also
|An3| ≤ 2
∫∫
f2
(
`0 +
v
an
)
K2
0 (z)W (v)
∣∣∣∣W (
v +
an
λn
z
)
−W (v)
∣∣∣∣ dz dv ≤
≤ c9
∫
K2
0 (z)ω1
(
an
λn
z
)
dz. (4)
The expression ω1(h) =
∫ ∣∣W (v + h) −W (v)
∣∣ dv is the L1-modulus of continuity of
the function W (x). It is evidently bounded as a function of h since ω1(h) ≤ 2‖W‖L1
.
Moreover, ω1(h) → 0 as h → 0. Therefore, by the Lebesque theorem on majorized
convergence, the integral in the right-hand part of (4) converges to zero as n→∞.
So, using this fact and (3) we obtain
(λnan)−1σ2
n(f) −→ σ2(f).
The lemma is proved.
Theorem 1. Let K(x) ∈ H, f(x) ∈ F, W (x) be bounded and W (x) ∈ L1(−∞,
∞). If λn →∞, an →∞, an/λn → 0 and
λna
2
n
n
→ 0 as n→∞, then
U
(1)
n −∆n(f)
σn(f)
d−→ N(0, 1),
where d denoted convergence in distribution, and N(a, σ) a random value having a
normal distribution with mean a and dispersion σ2.
Proof. We have
σ−1n (U (1)
n −∆n) =
√
n− 1
n
H(1)
n +H(2)
n ,
where
H(1)
n =
n∑
j=1
ξj ,
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 4
ON AN INTEGRAL SQUARE DEVIATION MEASURE WITH THE GENERALIZED WEIGHT . . . 519
H(2)
n =
1
nσn
n∑
j=1
(∫
α2
n(x,Xj)anW (an(x− `0)) dx−
−E
∫
α2
n(x−Xj)anW (an(x− `0)) dx
)
,
σ2
n ≡ σ2
n(f).
We will first establish the convergence of H(2)
n to zero in probability.
Indeed,
VarH(2)
n ≤ c10n−1σ−2n λ4nE
[ ∫
K2 (λn(x−X1)) anW (an(x− `0)) dx+
+
∫
(EK (λn(x−X1)))
2
anW (an(x− `0)) dx
]2
≤
≤ c11n−1σ−2n λ4n
{
E
[ ∫
K2 (λn(x−X1)) anW (an(x− `0)) dx
]2
+
+
[ ∫
(EK (λn(x−X1)))
2
anW (an(x− `0)) dx
]2}
=
= I(1)n + I(2)n ,
and also
I(1)n ≤ c12λ4nn−1σ−2n λ−2n a2n = c12
(
λnan
σ2
n
)
λnan
n
−→ 0,
I(2)n ≤ c13n−1σ−2n −→ 0.
Therefore,
VarH(2)
n = O
(
λnan
n
)
+O
(
1
nσ2
n
)
.
Hence H(2)
n
P−→ 0 as n→∞ (here and in the sequel the letter p above the arrow denote
convergence in probability).
To prove the assertion of Theorem 1 we need to show that H(1)
n
d−→ N(0, 1). To this
end, we use Theorem 4 from [6, p. 580] which contains the conditions of the central limit
theorem for sequences that form a difference-martingale. Let us show that our sequence
{ξk,Fk} satisfies these conditions. Note that
∑n
j=1
Eξ2j = 1 since, as can be easily
verified, Eξ2j = 2(j − 1)
[
n(n − 1)
]−1
. Asymptotic normality takes place if for every
ε ∈ (0, 1] and n→∞
n∑
k=1
E
[
ξ2kI (|ξk| ≥ ε)
∣∣∣Fk−1] P−→ 0
(the Lindeberg condition) and
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 4
520 E. NADARAYA, P. BABILUA, G. SOKHADZE
V 2
n =
n∑
k=1
E
(
ξ2k | Fk−1
) P−→ 1,
i.e., then
H(1)
n =
n∑
k=1
ξk
d−→ N(0, 1).
In the first place we verify that
V 2
n =
n∑
k=1
E
(
ξ2k | Fk−1
) P−→ 1 as n→∞. (5)
Taking the definition of ξj into account, we represent V 2
n in the form
V 2
n =
n∑
j=2
E
(
ξ2j | X1, . . . , Xj−1
)
=
=
n∑
j=2
E
(j−1∑
i=1
ηij
)2 ∣∣∣∣X1, . . . , Xj−1
=
=
n∑
j=2
E
(
j−1∑
i=1
η2ij
∣∣∣∣X1, . . . , Xj−1
)
+ 2
∑
i<`
E
(
ηijη`j
∣∣∣∣X1, . . . , Xj−1
)
=
=
n∑
j=2
E
(
j−1∑
i=1
η2ij
∣∣∣∣X1, . . . , Xj−1
)
+
+2
n∑
j=3
E
(
j−2∑
i=1
j−1∑
`=i+1
ηijη`j
∣∣∣∣X1, . . . , Xj−1
)
=
= Vn1 + Vn2.
Let us show that Vn1
P−→ 1 and Vn2
P−→ 0 as n→∞.
Denote
Φn(x, y) = EK (λn(x−X1))K (λn(y −X1))−
−EK (λn(x−X1))EK (λn(y −X1)) ,
εi = λ−2n
∫∫ [
αn(x,Xi)αn(y,Xi)Φn(x, y)Wn(x)Wn(y)
]
dx dy,
σn =
∫∫
Φ2
n(x, y)Wn(x)Wn(y) dx dy,
Zj =
j−1∑
i=1
(εi − σn).
Further, the definition of ηij implies that
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 4
ON AN INTEGRAL SQUARE DEVIATION MEASURE WITH THE GENERALIZED WEIGHT . . . 521
Vn1 =
4
n2σ2
n
n∑
j=2
E
(
j−1∑
i=1
∫∫
αn(x,Xi)αn(y,Xi)αn(x,Xj)αn(y,Xj)×
×Wn(x)Wn(y) dx dy
∣∣∣X1, . . . , Xj−1
)
=
=
4
n2σ2
n
n∑
j=2
j−1∑
i=1
∫∫
αn(x,Xi)αn(y,Xi)Eαn(x,Xj)αn(y,Xj)Wn(x)Wn(y) dx dy
and since
Eαn(x,Xi)αn(y,Xi) = λ2n
(
E (K (λn(x−X1))K (λn(y −X1)))−
−EK (λn(x−X1))EK (λn(y −X1))
)
we have
Vn1 =
4λ2n
n2σ2
n
n∑
j=2
j−1∑
i=1
∫∫
αn(x,Xi)αn(y,Xi)Φn(x, y)Wn(x)Wn(y) dx dy.
Therefore,
VarVn1 = E(Vn1 − EVn1)2 =
16λ8n
n4σ4
n
E
n∑
j=2
j−1∑
i=1
(εi − σn)
2
=
= Dn
n∑
j=2
E
(
j−1∑
i=1
(εi − σn)
)2
+ 2Dn
n−1∑
i=2
E (Zi(Zi+1 + . . .+ Zn)) ,
where
Dn =
16λ8n
n4σ4
n
.
It is obvious that
Zi+1 = Zi + (εi − σn),
Zi+2 = Zi + (εi − σn) + (εi+1 − σn),
..............................................................
Zn = Zi + (εi − σn) + . . .+ (εn−1 − σn).
Hence
E(Vn1 − EVn1)2 =
= Dn
n∑
j=2
E
(
j−1∑
i=1
(εi − σn)
)2
+ 2Dn
n−1∑
i=2
EZ2
i (n− i) =
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 4
522 E. NADARAYA, P. BABILUA, G. SOKHADZE
= Dn
n∑
j=2
(j − 1)E(ε1 − σn)2 + 2Dn
n−1∑
i=2
EZ2
i (n− i) = Bn1 +Bn2.
Let us estimate Bn1 and Bn2.
We have
Bn1 ≤ c14
λ8n
n2σ4
n
E(ε1 − σn)2. (6)
But
E(ε1 − σn)2 =
∫ {∫∫ [
K (λn(x− t))− EK (λn(x−X1))
]
×
×
[
K (λn(y − t))− EK (λn(y −X1))
]
×
×Φn(x, y)Wn(x)Wn(y) dx dy
}2
f(t) dt =
= c15
λ8n
n2σ4
n
(An1 +An2 +An3 +An4) , (7)
where
An1 =
∫ [ ∫∫
K (λn(x− t))K (λn(y − t)) Φn(x, y)×
×Wn(x)Wn(y) dx dy
]2
f(t) dt,
An2 =
∫ [ ∫∫
K (λn(x− t))EK (λn(y −X1)) Φn(x, y)×
×Wn(x)Wn(y) dx dy
]2
f(t) dt,
An3 =
∫ [ ∫∫
K (λn(y − t))EK (λn(x−X1)) Φn(x, y)×
×Wn(x)Wn(y) dx dy
]2
f(t) dt,
An4 =
∫ [ ∫∫
EK (λn(x−X1))EK (λn(y −X1)) Φn(x, y)×
×Wn(x)Wn(y) dx dy
]2
f(t) dt.
Since |Φn(x, y)| ≤ c16λ−1n , it can be easily established that
An1 ≤ c17
a4n
λ6n
, An2 ≤ c18
a2n
λ6n
,
An3 ≤ c18
a2n
λ6n
, An4 ≤ c19λ−6n .
(8)
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ON AN INTEGRAL SQUARE DEVIATION MEASURE WITH THE GENERALIZED WEIGHT . . . 523
Using (7) and (8) we obtain
E(ε1 − σn)2 ≤ c20
(
a4n
λ6n
+
a2n
λ6n
+
1
λ6n
)
. (9)
This and (6) imply
Bn1 ≤ c21
(
λnan
σ2
n
)2
a2n
n2
−→ 0. (10)
Further, it is not difficult to see that
Bn2 =
32λ8n
n4σ4
n
n−1∑
i=2
EZ2
i (n− i) =
32λ8n
n4σ4
n
n−1∑
i=2
(n− i)(i− 1)E(ε1 − σn)2.
This and (9) imply that
Bn2 ≤ c22
λ2na
4
n
nσ4
n
= c22
(
λnan
σ2
n
)2
a2n
n
−→ 0.
Therefore,
VarVn1 = E(Vn1 − EVn1)2 −→ 0.
On the other hand,
EVn1 =
4λ2n
n2σ2
n
n∑
j=2
j−1∑
i=1
∫∫
Eαn(x,Xi)αn(y,Xi)Φn(x, y)Wn(x)Wn(y) dx dy =
=
4
n2σ2
n
n∑
j=2
j−1∑
i=1
∫∫
(Eαn(x,X1)αn(y,X1))
2
Wn(x)Wn(y) dx dy =
=
n2 − n
n2
= 1− 1
n
−→ 1.
Therefore,
Vn1
P−→ 1.
Now let consider Vn2 and show that Vn2
P−→ 0. Taking the inequality
E
(
m∑
i=1
Zi
)2
≤
(
n∑
i=1
(EZ2
i )1/2
)2
,
into account, we obtain
EV 2
n2 = DnE
(
n∑
j=3
j−2∑
i=1
j−1∑
`=i+1
∫∫
αn(x,Xi)αn(y,X`)Φn(x, y)×
×Wn(x)Wn(y) dx dy
)2
=
= DnE
(
n∑
j=3
∫∫ j−2∑
i=1
αn(x,Xi)gi(y)Φn(x, y)Wn(x)Wn(y) dx dy
)2
≤
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 4
524 E. NADARAYA, P. BABILUA, G. SOKHADZE
≤ Dn
[
n∑
j=3
{
E
(∫∫ j−1∑
i=1
αn(x,Xi)gi(y)Φn(x, y)×
×Wn(x)Wn(y) dx dy
)2}1/2]2
, (11)
where
Dn =
16λ8n
n4σ4
n
,
αn(x,Xi) = K (λn(x−Xi))− EK (λn(x−X1)) ,
gi(y) =
j−1∑
`=i+1
αn(y,X`).
Since
E αn(x,Xi)gi(y)αn(y,Xr)gr(y) = 0 as i < r,
(11) takes the form
EV 2
n2 =
= Dn
[
n∑
j=3
{
j−2∑
i=1
E
(∫∫
αn(x,Xi)gi(y)Φn(x, y)Wn(x)Wn(y) dx dy
)2}1/2]2
.
(12)
Next, elementary calculations show that
E
j−2∑
i=1
(∫∫
αn(x,Xi)gi(y)Φn(x, y)Wn(x)Wn(y) dx dy
)2
=
=
j−2∑
i=1
E
∫∫∫∫
αn(x1, Xi)αn(x2, Xi)×
×
j−1∑
`1=i+1
j−1∑
`2=i+1
αn(y1, X`1)αn(y2, X`2)Φn(x1, y1)Φn(x2, y2)×
×Wn(x1)Wn(x2)Wn(y1)Wn(y2) dx1 dx2 dy1 dy2 =
=
j−2∑
i=1
∫∫∫∫
E αn(x1, Xi)αn(x2, Xi)×
×
j−1∑
`1=i+1
E αn(y1, X`)αn(y2, X`)Φn(x1, y1)Φn(x2, y2)×
×Wn(x1)Wn(x2)Wn(y1)Wn(y2) dx1 dx2 dy1 dy2 =
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 4
ON AN INTEGRAL SQUARE DEVIATION MEASURE WITH THE GENERALIZED WEIGHT . . . 525
= O
(
j2
∫∫∫∫ ∣∣∣E αn(x1, X1)αn(x2, X1)E αn(y1, X1)αn(y2, X1)
∣∣∣×
×Wn(x1)Wn(x2)Wn(y1)Wn(y2) dx1 dx2 dy1 dy2
)
. (13)
Recalling the definition of αn(x, y) and performing the change of variables, we obtain∫∫∫∫ ∣∣∣E αn(x1, X1)αn(x2, X1)E αn(y1, X1)αn(y2, X1)
∣∣∣×
×Wn(x1)Wn(x2)Wn(y1)Wn(y2) dx1 dx2 dy1 dy2 ≤
≤ c23λ−7n
∫∫∫∫ [ ∫
K(w1)K(z1 − w1)f
(
x1 −
w1
λn
)
dw1+
+λnEK (λn(x1 −X1))EK (λn(x1 −X1) + z1)
]
×
×
[ ∫
K(w2)K(z2 − z3 − w2)f
(
x1 +
z2
λn
− w2
λn
)
dw2+
+λnEK (λn(x1 −X1) + z2)EK (λn(x1 −X1) + z3)
]
×
×
[ ∫
K(w3)K(z2 − w3)f
(
x1 −
w3
λn
)
dw3+
+λnEK (λn(x1 −X1))EK (λn(x1 −X1) + z2)
]
×
×
[ ∫
K(w4)K(z3 − z1 − w4)f
(
x1 +
z1
λn
− w4
λn
)
dw4+
+λnEK (λn(x1 −X1) + z1)EK (λn(x1 −X1) + z3)
]
×
×a4nW (an(x1 − `0))W
((
x1 +
z1
λn
− `0
)
an
)
×
×W
(
an
(
x1 +
z2
λn
− `0
))
W
(
an
(
x1 +
z3
λn
− `0
))
dx1 dz1 dz2 dz3 ≤
≤ c24λ−7n a3n, (14)
since the integrals contained in (12) are majorised by the value a3n.
Thus, by virtue of (13) and (14), from (12) it follows that
EV 2
n2 ≤ c25
(
λnan
σ2
n
)2
an
λn
−→ 0.
Therefore,
n∑
j=1
E
(
ξ2j | Fk−1
)
= Vn1 + Vn2
P−→ 0. (15)
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 4
526 E. NADARAYA, P. BABILUA, G. SOKHADZE
Let us now proceed to establishing the validity of the Lindeberg condition
n∑
k=1
E
[
ξ2kI (|ξk| ≥ ε) | Fk−1
]
P−→ 0. (16)
For (16) to be valid it suffices to show that
n∑
j=1
Eξ4j −→ 0 as n→∞. (17)
Indeed,
P
{
n∑
k=1
E
[
ξ2kI (|ξk| ≥ ε)
∣∣∣Fk−1] ≥ δ} ≤
≤ δ−1
n∑
j=1
E
(
E
[
ξ2j (I(ξj ≥ ε))
∣∣∣Fk−1]) = δ−1
n∑
j=1
E
[
ξ2j (I(ξj ≥ ε))
]
≤
≤ δ−1ε−2
n∑
j=1
E
(
ξ4j I(ξj ≥ ε)
)
≤ δ−1ε−2
4∑
j=1
Eξ4j .
We will prove (17). By the definitions of ηik and ξk, we obtain
n∑
k=1
Eξ4k =
16
n4σ4
n
(
M (1)
n +M (2)
n
)
, (18)
where
M (1)
n =
n∑
k=1
(k − 1)
∫ [
E
4∏
i=1
αn(xi, X1)
]4 4∏
i=1
Wn(xi) dx,
M (2)
n = 3
n∑
k=1
(k − 1)(k − 2)
∫ [
E
1∏
i=0
αn(xi+1, X1)αn(xi+3, X2)×
×E
4∏
i=1
αn(xi, X1)Wn(xi)
]
dx, dx = dx1 . . . dx4.
Let us estimate M (1)
n and M (2)
n . We have
M (1)
n =
=
n∑
k=1
(k − 1)
∫∫ (∫
αn(x1, u)αn(x1, v)Wn(x1) dx1
)4
f(u)f(v) du dv ≤
≤ c26n2
∫∫ [
δ4(u, v) +
(∫
f(t)δ(u, t) dt
)4
+
+
(∫
f(t)δ(v, t) dt
)4
+
(∫∫
δ(x, y)f(x)f(y) dx dy
)4]
f(u)f(v) du dv,
(19)
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 4
ON AN INTEGRAL SQUARE DEVIATION MEASURE WITH THE GENERALIZED WEIGHT . . . 527
where
δ(x, y) = λ2n
∫
K (λn(x− u))K (λn(x− v))Wn(x) dx.
Since
sup
v
∫
δs(u, v)f(u) du ≤ c27asnλsn sup
v
∫
Ks
0 (λn(u− v)) du ≤ c28asnλs−1n ,
s = 2, 4,
formula (19) implies
M (1)
n ≤ c29n2a4nλ3n, (20)
and also
M (2)
n = 3
n∑
k=1
(k − 1)(k − 2)
∫∫∫∫ (∫
αn(x1, u)αn(x1, t)Wn(x1) dx1
)2
×
×
(∫
αn(x2, v)αn(x2, t)Wn(x2) dx2
)2
f(u)f(v)f(t) du dv dt ≤
≤ 3
n∑
k=1
(k − 1)(k − 2)
∫∫∫ (
δ2n(u, t) +O(a4nλn)
)
×
×
(
δ2n(v, t) +O(λna
4
n)
)
f(u)f(v)f(t) du dv dt ≤
≤ c30n3a4nλ3n. (21)
From (18), (20) and (21) it follows that
n∑
k=1
Eξ4k ≤ c31
λ3na
4
n
nσ4
n
= c31
(
anλn
σ2
n
)2
a2nλn
n
.
Therefore,
lim
n→∞
n∑
k=1
Eξ4k = 0.
The theorem is proved.
Theorem 2. Let K(x) ∈ H, f(x) ∈ F, W (x) be bounded and W (x) ∈ L1(−∞,
∞). If λn →∞, an →∞,
an
λn
→ 0 and
λna
2
n
n
→ 0 as n→∞, then
(λnan)−1/2σ−1(f)
(
U (1)
n −∆n(f)
)
d−→ N(0, 1),
where
σ2(f) = 2f2(`0)
∫
K2
0 (z) dz
∫
W 2(v) dv, f(`0) 6= 0.
Proof. Follows from Lemma 2 and Theorem 1.
Theorem 3. Let K(x) ∈ H, f(x) ∈ F, W (x) be bounded, W (−x) = W (x),
x ∈ R and x2W (x) ∈ L1(R). If λn → ∞, an → ∞,
an
λn
→ 0,
λna
2
n
n
→ 0 and
λna
−5
n → 0, then
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 4
528 E. NADARAYA, P. BABILUA, G. SOKHADZE
(
λn
an
)1/2
σ−1(f)
(
U (2)
n −∆(f)
)
d−→ N(0, 1),
where
∆(f) = f(`0)
∫
K2(u) du
∫
W (x) dx, U (2)
n = λ−1n U (1)
n .
Proof. We have
∆n(f) = n
∫
E (fn(x)− Efn(x))
2
Wn(x) dx =
= λn
∫∫
K2(u)f
(
x− u
λn
)
anW (an(x− `0)) du dx−
−
∫ (∫
K(v)f
(
x− v
λn
)
dv
)2
anW (an(x− `0)) dx. (22)
Since
λn
∫∫
K2(u)f
(
x− u
λn
)
anW (an(x− `0)) du dx =
= λn
∫
K2(u) du
∫
f
(
`0 +
v
an
)
W (v) dv +O(1) =
= f(`0)λn
∫
K2(u) du
∫
W (u) du+O
(
λn
a2n
)
+O(1)
and ∫ (∫
K(v)f
(
x− v
λn
)
dv
)2
anW (an(x− `0)) dx = O(1),
from these formulas and (22) we obtain
∆n(f) = λn
[
f(`0)
∫
K2(u) du
∫
W (v) dv +O
(
1
a2n
)
+O
(
1
λn
)]
=
= λn
[
∆(f) +O(a−2n ) +O(λ−1n )
]
.
Therefore,
(λnan)−1/2σ−1
(
U (1)
n −∆n
)
−
√
λn
an
σ−1
(
U (2)
n −∆
)
=
= O
( √
λn√
an a2n
)
+O
((
1
λnan
)1/2)
.
Since the right-hand part of the latter equality tends to zero by virtue of the condition
λn/a
−5
n → 0.
The theorem is proved.
The case where in U (2)
n Efn(x) is replaced by f(x) is more natural for applications.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 4
ON AN INTEGRAL SQUARE DEVIATION MEASURE WITH THE GENERALIZED WEIGHT . . . 529
Theorem 4. Let K(x), f(x) and W (x) satisfy the conditions of Theorem 3. If
λn →∞, an →∞,
an
λn
→ 0,
λna
2
n
n
→ 0,
λn
a5n
→ 0,
√
nan λ
−5/2
n → 0 and
n
√
an
λ−9/2n → 0,
then (
λn
an
)1/2
σ−1(f) (Un −∆(f))
d−→ N(0, 1),
Un =
n
λn
∫
(fn(x)− f(x))
2
Wn(x) dx.
Proof. We have√
λn
an
(
Un − U (2)
n
)
=
√
λn
an
Θn +
√
λn
an
Rn,
where
Θn =
n
λn
∫
(Efn(x)− f(x))
2
Wn(x) dx,
Rn = 2
n
λn
∫
(fn(x)− Efn(x)) (Efn(x)− f(x))Wn(x) dx.
Let us estimate
√
λn/anE|Rn|. Since
cov (fn(x), fn(y)) = n−1λ2n
{∫
K (λn(x− u))K (λn(y − u)) f(u) du−
−
∫
K (λn(x− u)) f(u) du
∫
K (λn(y − u)) f(u) du
}
,
we obtain √
λn
an
E|Rn| ≤
≤ 2
n√
anλn
{
λ2n
n
E
[ ∫
K (λn(x−X1)) (Efn(x)− f(x))Wn(x) dx
]2
−
−λ
2
n
n
[ ∫
EK (λn(x−X1)) (Efn(x)− f(x))Wn(x) dx
]2}1/2
≤
≤ 2
n√
anλn
{
λ2n
n
∫
f(u) du
[ ∫
K (λn(x− u)) (Efn(x)− f(x))Wn(x) dx
]2}1/2
.
(23)
Further, since
Efn(x)− f(x) = O(λ−2n ),
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 4
530 E. NADARAYA, P. BABILUA, G. SOKHADZE
uniformly with respect to x ∈ R = (−∞,∞), from (23) we obtain√
λn
an
E|Rn| ≤ c32
√
nan λ
−5/2
n −→ 0,
and also √
λn
an
Θn ≤ c33
n
√
an
λ−9/2n −→ 0.
The theorem is proved.
The conditions of Theorem 4 for λn and an are fulfilled, for instance, if it is assumed
that λn = n1/2+ε and an = nε, 1/8 < ε < 1/6.
2. Asymptotic power of the goodness-of-fit test based on Un. The assertion
of Theorem 4 enables us to construct tests of an asymptotic level α for verifying the
hypothesis H0 by which f(x) = f0(x) and f0(`0) 6= 0. To this end, we should calculate
Un and discard H0 if
Un ≥ dn(α) = ∆(f0) +
(
λn
an
)−1/2
εασ(f0), (24)
where
∆(f0) = f0(`0)
∫
K2(u) du
∫
W (x) dx,
σ2(f0) = 2f20 (`0)
∫
K2
0 (z) dz
∫
W 2(v) dv,
K0 = K ∗K,
εα is the quantile of the level α of the standard normal distribution Φ(x).
Now we will investigate the asymptotic behavior of test (24) or, more exactly, the
behavior of the power function for n→∞.
Let us consider the question whether test (24) is consistent.
The following statement is true.
Theorem 5. Let the conditions of Theorem 4 be fulfilled. Then for n→∞
Πn(f1) = PH1
{
Un ≥ dn(α)
}
−→ 1.
Therefore the test defined in (24) is consistent against any alternativeH1 : f(x) = f1(x),
f1(x) 6= f0(x) on the set of a positive Lebesgue measure and f1(`0) 6= f0(`0).
Proof. It is easy to see that
Πn(f1) =
= PH1
{
n
λn
∫
(fn(x)− f0(x))
2
Wn(x) dx ≥ ∆(f0) +
(
λn
an
)−1/2
σ(f0)εα
}
=
= PH1
{
n
λn
∫
(fn(x)− f1(x))
2
Wn(x) dx ≥
≥ ∆(f0) +
(
λn
an
)−1/2
σ(f0)εα −
n
λn
Rn−
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ON AN INTEGRAL SQUARE DEVIATION MEASURE WITH THE GENERALIZED WEIGHT . . . 531
−2
∫
(fn(x)− f1(x))Wn(x)ψn(x) dx
n
λn
}
=
= PH1
{√
λn
an
σ−1(f1) (U∗n −∆(f1)) ≥
≥
√
λn
an
[
∆(f0)−∆(f1)
]
σ−1(f1) +
σ(f0)
σ(f1)
εα −
n√
λnan
σ−1(f1)Rn−
−2
n√
λnan
∫
(fn(x)− f1(x))Wn(x)ψn(x) dxσ−1(f1)
}
=
= PH1
{√
λn
an
σ−1(f1) (U∗n −∆(f1)) ≥
≥ − n√
λnan
[
σ−1(f1)Rn +
λn
n
(∆(f1)−∆(f0))σ−1(f1)+
+2σ−1(f1)
∫
(fn(x)− f1(x))ψn(x)Wn(x) dx+
√
λnan
n
σ(f0)σ−1(f1)εα
]}
, (25)
where
U∗n =
n
λn
∫
(fn(x)− f1(x))
2
Wn(x) dx,
ψn(x) = (f1(x)− f0(x))Wn(x), Wn(x) = anW (an(x− `0)) ,
Rn =
∫
(f1(x)− f0(x))
2
Wn(x) dx.
Furthermore, using the inequalities
E
(∫
(fn(x)− f1(x))
2 (
f21 (x) + f20 (x)
)
dx
)
≤ c34
λn
n
+ c35 λ
−4
n
and ∫
W 2
n(x) dx ≤ c36an,
we can state that ∫
(fn(x)− f1(x))ψn(x) dx
P−→ 0.
Therefore, (25) implies
Πn(f1) = PH1
{√
λn
an
σ−1(f1) (U∗n −∆(f1)) ≥
≥ − n√
λnan
(
σ−1(f1)Rn + op(1)
)}
. (26)
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 4
532 E. NADARAYA, P. BABILUA, G. SOKHADZE
Since
√
λn
an
σ−1(f1)(U∗n − ∆(f1)) is distributes assimptotically normally to (0, 1)
in the case of the hypothesis H1,
n√
λnan
−→∞
and
Rn −→ (f1(`0)− f0(`0))
2
∫
W (u) du > 0 as n→∞,
from (26) it follows that Πn(f1)→ 1.
The theorem is proved.
Thus the power of test (24) for any fixed alternative tends to 1 as n → ∞.
Nevertheless more profound properties of the test are revealed when investigating the
question how the test reacts to “small” deviations from the verified hypothesis, i.e.,
when instead of the fixed alternative we consider the sequence of alternatives {H1n}
approaching with the basic hypothesis H0 as n → ∞. Let us consider the sequence of
alternatives of the form [2, 3]
H1 : f1(x) = f0(x) + αnϕ
(
x− `n
γn
)
+ o(αnγn), `n = `0 + o(γn),
where αn ↓ 0, γn ↓ 0, the function ϕ(x) ∈ F and
∫
ϕ(x) dx = 0.
Theorem 6. LetK(x), f1(x), W (x), λn and an satisfy the conditions of Theorem 4.
If, in addition to this,
W (0) 6= 0, αnγn = o(n−1/2),
nλ−1/2n a1/2n γnα
2
n −→ γ0, λna
−1
n α2
n −→ 0, anγn −→ 0,
√
nαn
√
an λ
−5/2
n γ−2n −→ 0, λ4nγ
5
nan −→∞,
and
anλnαn −→∞ as n→∞,
then
PH1
{
Un ≥ dn(α)
}
−→ 1− Φ
(
εα −
γ0W (0)
σ(f0)
∫
ϕ2(x) dx
)
.
Proof. We write Un as a sum
Un = U (2)
n +An1 +An2,
U (2)
n =
n
λn
∫
(fn(x)− Efn(x))
2
Wn(x) dx,
An1 =
n
λn
∫
(Efn(x)− f0(x))
2
Wn(x) dx,
An2 = 2
n
λn
∫
(fn(x)− Efn(x)) (Efn(x)− f0(x))Wn(x) dx,
where E(· ) is the mathematical expectation under the hypothesis H1. Therefore, we
obtain
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ON AN INTEGRAL SQUARE DEVIATION MEASURE WITH THE GENERALIZED WEIGHT . . . 533
PH1
{
Un ≥ dn(α)
}
= PH1
{
1√
anλn
σ−1(f1)
[
U (1)
n − EU (1)
n
]
≥
≥ σ(f0)
σ(f1)
εα +
√
λ
an
σ−1(f1)
[
λ−1n ∆n(f1)−∆(f0)
]
−
−
√
λn
an
σ−1(f1)An1 +
√
λn
an
σ−1(f1)An2
}
. (27)
Tracing the proofs of Theorems 1 and 2, it is not difficult to make sure that
1√
anλn
σ−1(f1)
[
U (1)
n − EU (1)
n
]
d−→ N(0, 1).
Let us show that √
λn
an
σ−1(f1)An2
P−→ 0.
Indeed, √
λn
an
σ−1(f1)E|An2| ≤
≤ c1
1√
anλn
{
λ2n
n
∫
f1(u) du
[ ∫
K (λn(x− u)) (Efn(x)− f1(x))Wn(x) dx
]2}1/2
and also
Efn(x) = f1(x) +O(λ−2n ) +O
(
αn
λ2nγ
2
n
)
.
Hence √
λn
an
σ−1(f1)E|An2| = O
((
nan
λ5n
)1/2)
+O
(√
nαn
√
an
λ
5/2
n γ2n
)
.
Therefore, √
λn
an
σ−1(f1)An2
P−→ 0. (28)
Next, by using the condition nλ−1/2n a
1/2
n γnα
2
n −→ γ0 it is not difficult to establish that√
λn
an
An1 =
nα2
n√
λnan
∫
ϕ2
(
x− `n
γn
)
Wn(x) dx+O(na−1/2n λ−9/2n )+
+O(λ−4n γ−5n a−1n ) +O
(
1
αnλ2n
)
+O
(
1
λnγn
)
+O(a−1n α−1n λ−1n ).
From this and the Lebesgue theorem on majorized convergence it follows that√
λn
an
σ−1(f1)An1 =
= σ−1(f1)nλ−1/2n a1/2n α2
nγn
1
anγn
∫
ϕ2
(
t
anγn
)
W (t) dt+
+O
(
na−1/2n λ−9/2n
)
+O
(
λ−4n γ−5n a−1n
)
+O
(
1
αnλ2n
)
+O
(
1
λnγn
)
+
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 4
534 E. NADARAYA, P. BABILUA, G. SOKHADZE
+O
(
a−1n α−1n λ−1n
)
−→ γ0W (0)
σ(f0)
∫
ϕ2(u) du. (29)
Finally, we can easily show that√
λn
an
σ−1(f1)
[
λ−1n ∆n(f1)−∆(f0)
]
= O
((
λn
a5n
)1/2)
+O
(
λnα
2
n
an
)
. (30)
Thus (27) – (30) imply
PH1
{
Un ≥ dn(α)
}
−→ 1− Φ
(
εα −
γ0(W0)
σ(f0)
∫
ϕ2(u) du
)
.
The theorem is proved.
It is well known that the limit power of the Rosenblatt – Bickel test [1 – 3]
Tn ≥
∫
f0(x)W (x) dx
∫
K2(u) du+ λ−1/2n εασ0,
Tn =
n
λn
∫
(fn(x)− f0(x))
2
W (x) dx, (31)
σ2
0 = 2
∫
f20 (x)W 2(x) dx
∫
K2
0 (u) du.
For verifying the hypothesis H0 : f(x) = f0(x) against the alternative
H1 : f1(x) = f0(x) + αnϕ
(
x− `n
γn
)
+ o(αnγn), `n = `0 + o(γn),
where λn = nδ, αn = n−α, γn = n−β for some α, β and δ, for which α + β > 1/2,
1− 2α − β = δ/2 (for example, α = 9/35, β = 2/7, δ = 2/5 or α = 1/6, β = 5/12,
δ = 1/2), is equal to,
γ(T ) = 1− Φ
(
εα −
W (`0)
σ0
∫
ϕ2(u) du
)
,
while the limit power γ(U) of test (24) is equal to 1 (see (29)) for an = nε, ε < δ.
However, for some α, β, δ and ε, for which α+ β > 1/2, 1− 2α− β + ε/2 = δ/2
(for example, α = 9/35, β = 2/7, ε = 1/6, δ = 17/30), the limit power of test (24),
by Theorem 6, is equal to
γ(U) = 1− Φ
(
εα −
W (0)
σ(f0)
∫
ϕ2(u) du
)
,
while the limit power γ(T ) of test (31) is equal to 1−Φ(εα). Therefore, choosing between
(31) and (24), we will prefer the test based on Un.Moreover, for weight functionsW (x),
for which W (0) > 0, the goodness-of-fit test of the hypothesis H0 against alternatives
of form H1 are asymptotically strictly unbiased since the mathematical expectation
γ0W (0)
∫
ϕ2(u) du > 0 is equal to zero if and only if ϕ(x) = 0, x ∈ (−∞,∞).
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 4
ON AN INTEGRAL SQUARE DEVIATION MEASURE WITH THE GENERALIZED WEIGHT . . . 535
1. Bickel P. J., Rosenblatt M. On some global measures of the deviations of density function estimates //
Ann. Statist. – 1973. – 1. – P. 1071 – 1095.
2. Rosenblatt M. A quadratic measure of deviation of two-dimensional density estimates and a test of
independence // Ibid. – 1975. – 3. – P. 1 – 14.
3. Nadaraya E. A. Nonparametric estimation of probability densities and regression curves. – Dordrecht:
Kluwer Acad. Publ. Group, 1989.
4. Hall P. Central limit theorem for integrated square error of multivariate nonparametric density estimators
// J. Multivar. Anal. – 1984. – 14, № 1. – P. 1 – 16.
5. Cai T. Tony, Low Mark G. Nonparametric estimation over shrinking neighborhoods: superefficiency and
adaptation // Ann. Statist. – 2005. – 33, № 1. – P. 184 – 213.
6. Shiryaev A. N. Probability (in Russian). – Moscow: Nauka, 1989.
Received 03.03.09
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 4
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