Statistics & Operations Research TransactionsSORT 29 (2) JulyDecember 2005, 201216
Statistics &Operations ResearchTransactions
On the role played by the ﬁxed bandwidth in theBickelRosenblatt goodnessofﬁt test
c
Institut d’Estad´ıstica de Catalunyasort@idescat.esISSN: 16962281www.idescat.net
/
sort
Carlos Tenreiro
Universidade de Coimbra
Abstract
For the BickelRosenblatt goodnessofﬁt test with ﬁxed bandwidth studied by Fan (1998) we deriveits Bahadur exact slopes in a neighbourhood of a simple hypothesis
f
=
f
0
and we use them to get abetter understanding on the role played by the smoothing parameter in the detection of departuresfrom the null hypothesis. When
f
0
is a univariate normal distribution and we take for kernel the standardnormal density function, we compute these slopes for a set of Edgeworth alternatives which give us adescription of the test properties in terms of the bandwidth
h
. A simulation study is presented whichindicates that ﬁnite sample properties are in good accordance with the theoretical properties based onBahadur local efﬁciency. Comparisons with the quadratic classical EDF tests lead us to recommenda test based on a combination of bandwidths in alternative to AndersonDarling or Cram´ervon Misestests.
MSC:
62G10, 62G20
Keywords:
goodnessofﬁt test, kernel density estimator, Bahadur efﬁciency.
1 Introduction
Let
X
1
,
X
2
,...,
X
n
,...
be a sequence of independent and identically distributed
d
dimensional random vectors with unknown density function
f
. As it has been shownby Bickel and Rosenblatt (1973), a test of the simple hypothesis
H
0
:
f
=
f
0
against thealternative
H
a
:
f
f
0
, where
f
0
is a ﬁxed density function on
R
d
, can be based on the
L
2
distance between the kernel density estimator of
f
introduced by Rosenblatt (1956)
Address for correspondence:
Carlos Tenreiro. Departamento de Matem´atica, Universidade de Coimbra,Apartado 3008, 3001454 Coimbra, Portugal. Phone: (351) 239 791 155. Fax: (351) 239 832 568. Email:tenreiro@mat.uc.ptReceived: July 2004Accepted: April 2005
202
On the role played by the ﬁxed bandwidth in the BickelRosenblatt goodnessofﬁt test
and Parzen (1962), and its mathematical expectation under the null hypothesis (see alsoFan (1994) and Gouri´eroux and Tenreiro (2001)):
I
2
n
(
h
n
)
=
n
{
f
n
(
x
)
−
E
0
f
n
(
x
)
}
2
dx
,
(1)where, for
x
∈
R
d
,
f
n
(
x
)
=
1
n
n
i
=
1
K
h
n
(
x
−
X
i
)
,
K
h
n
=
K
(
·
/
h
n
)
/
h
d n
with
K
a kernel, that is, a bounded and integrable function on
R
d
, and(
h
n
) is a sequence of strictly positive real numbers converging to zero, when
n
goes toinﬁnity (bandwidth). The BickelRosenblatt test is asymptotically consistent and has anormal asymptotic distribution under the null hypothesis.Following an idea of Anderson, Hall and Titterington (1994) that have used kerneldensity estimators with ﬁxed bandwidth for testing the equality of two multivariateprobability density functions, Fan (1998) uses the statistic (1) with a constant bandwidthfor testing the composite hypothesis that
f
is a member of a general parametricfamily of density functions. He provides an alternative asymptotic approximation forthe ﬁnitesample properties of the BickelRosenblatt test by showing that, for a ﬁxed
h
, the asymptotic distribution of
I
2
n
(
h
) is an inﬁnite sum of weighted
χ
2
randomvariables. Moreover, Fan (1998) proves that
I
2
n
(
h
) can be interpreted as a
L
2
weighteddistance between the empirical characteristic function and the parametric estimate of the characteristic function implied by the null model with weight function
t
→
φ
K
(
th
)

2
.In the important case of testing univariate or multivariate normality, and taking for
K
the standard normal density function, the role played by
h
in the power performance of the test is assessed in simulation studies by Epps and Pulley (1983), Henze and Zirkler(1990) and Henze and Wagner (1997).Restricting our attention to the test of a simple hypothesis, the main purpose of thispaper is to derive the Bahadur local exact slopes of goodnessofﬁt tests based on
I
2
n
(
h
),for a ﬁxed
h
>
0, and use them to get a better understanding of the role played bythe smoothing parameter in the detection of departures from the null hypothesis. Forcompleteness reasons we give in Section 2 the asymptotic null distribution and theconsistency of the test based on kernel density estimators with a ﬁxed bandwidth. Usingthe integral and quadratic form of
I
2
n
(
h
), we derive in Section 3 its Bahadur local exactslopes. They naturally depend on the smoothing parameter, on the kernel, on the nulldensity
f
0
and, ﬁnally, on the considered departure direction from the null hypothesis. InSection 4, in the particular case of a test for a simple univariate hypothesis of normalityand taking for
K
the standard normal density function, the Bahadur local slopes arenumerically evaluated for di
ﬀ
erent values of
h
for a set of Edgeworth alternatives. Thesealternatives express departures from the null hypothesis in terms of each one of the ﬁrst
Carlos Tenreiro
203
four moments. The tests based on
I
2
n
(
h
) for di
ﬀ
erent values of
h
are compared with thecorresponding ones of the quadratic EDF tests of AndersonDarling (
A
2
) and Cram´ervon Mises (
W
2
). The results we obtain suggest that a large bandwidth is adequate fordetection of location alternatives whereas a small bandwidth is adequate for detectionof alternatives for scale, skewness and kurtosis. A simulation study indicating that ﬁnitesample propertiesof tests
I
2
are in good accordance with the theoretical propertiesbasedon the Bahadur local slopes is also presented. Moreover, if one does not know muchabout the unknown density function it suggests that a test based on a combination of bandwidths,thatestablishacompromisebetweenthetwooppositee
ﬀ
ectsthatthechoiceof
h
has in the detection of location and nonlocation alternatives, is a good practicalrecommendation in alternative to traditional
A
2
or
W
2
tests.For convenience of presentation the proofs of some results in this article are givenin Section 5. We denote by
asn
→
+
∞
−→
the convergence with probability 1 and by
d n
→
+
∞
−→
theconvergence in distribution.
2 Asymptotic null distribution and consistency
Consider the following assumptions on
K
which ensure that
d
(
f
,
g
)
=
{
K
h
f
(
x
)
−
K
h
g
(
x
)
}
2
dx
1
/
2
, where
denotes the convolution product, is a distance on the set of integrable functions (see Anderson
et al.
(1994)).
Assumptions on
K
(K)
K
is a bounded and integrable function on
R
d
with Fourier transform
φ
K
such that
{
t
∈
R
d
:
φ
K
(
t
)
=
0
}
has Lebesgue measure zero.In order to derive the asymptotic distribution of
I
2
n
(
h
) under
H
0
for a ﬁxed
h
>
0, weﬁrst note that
I
2
n
(
h
) is a
V
statistic, that is,
I
2
n
(
h
)
=
1
n
n
i
,
j
=
1
Q
h
(
X
i
,
X
j
)
,
(2)with kernel
Q
h
(
u
,
v
)
=
k
(
x
,
u
;
h
)
k
(
x
,
v
;
h
)
dx
,
where
k
(
x
,
u
;
h
)
=
K
h
(
x
−
u
)
−
K
h
f
0
(
x
)
,
(3)for
u
,
v
,
x
∈
R
d
. From the hypothesis on
K
, the kernel
Q
h
is bounded. Therefore the
204
On the role played by the ﬁxed bandwidth in the BickelRosenblatt goodnessofﬁt test
functions
u
→
Q
h
(
u
,
u
) and
Q
h
are
P
0
and
P
0
⊗
P
0
integrable, respectively, where
P
0
=
f
0
λ
and
λ
istheLebesguemeasurein
B
(
R
d
).Moreover,
Q
h
issymmetricanddegenerate,i.e.,
Q
h
(
·
,
v
)
dP
0
(
v
)
=
0
,
a
.
e
.
(
P
0
)
.
From Gregory (1977), we know that the asymptoticdistribution of
I
2
n
(
h
) under
H
0
can be characterized in terms of the eigenvalues of thesymmetric HilbertSchmidt operator
A
h
deﬁned, for
q
∈
L
2
(
R
d
,
B
(
R
d
)
,
P
0
)
=
:
L
2
(
P
0
),by(
A
h
q
)(
u
)
=
Q
h
(
u
,
v
)
q
(
v
)
dP
0
(
v
)
.
(4)In view of the degeneracy property of
Q
h
,
q
0
,
h
=
1 is an eigenfunction of
A
h
corresponding to the eigenvalue
λ
0
,
h
=
0. Denoting by
1
the subspace generatedby
q
0
,
h
and
H
(
P
0
)
=
g
∈
L
2
(
P
0
) :
gdP
0
=
0
the tangent space of
P
0
, we have
L
2
(
P
0
)
=
1
⊕
H
(
P
0
). The operator
A
h
is positive deﬁnite on
H
(
P
0
) as follows from theintegral form (3) of
Q
h
and assumption (K). In fact, if
A
h
q
,
q
=
0, for some
q
∈
H
(
P
0
),where
·
,
·
denotes the usual inner product in
L
2
(
P
0
), we have0
=
q
(
u
)
k
(
·
,
u
;
h
)
dP
0
(
u
)
=
K
(
qf
0
)(
·
)
,
a
.
e
.
(
λ
)
,
yielding
φ
K
(
t
)
φ
qf
0
(
t
)
=
0, for all
t
∈
R
d
.
From assumption (K) and the continuity of the Fourier transform, we deduce that
φ
qf
0
(
t
)
=
0,
t
∈
R
d
, i.e.,
q
=
0
,
a
.
e
.
(
P
0
).Finally, using the the inﬁnitedimensionality of
H
(
P
0
) and the positivity of
A
h
on
H
(
P
0
) we can conclude that
A
h
has a countable inﬁnite collection
{
λ
k
,
h
,
k
∈
N
}
of strictlypositive eigenvalues (see Dunford and Schwartz (1963), Corollary X.4.5).The following result follows from the limit distribution of degenerate Vstatistics(cf. Theorem 4.3.2 of Koroljuk and Borovskich (1989)). Remark that the asymptoticdistribution presented by Fan (1998) in Theorem 4.2, is not correct. In general the
P
0
integrability of
u
→
Q
h
(
u
,
u
) is not a su
ﬃ
cient condition for
λ
k
,
h
<
∞
.
Theorem 1
If assumption
(K)
is fulﬁlled then, under H
0
we have I
2
n
(
h
)
d n
→
+
∞
−→
I
∞
,
with I
∞
=
Q
h
(
u
,
u
)
dP
0
(
u
)
+
∞
k
=
1
λ
k
,
h
(
Z
2
k
−
1)
,
where the sequence
(
λ
k
,
h
)
, with
λ
1
,
h
≥
λ
2
,
h
≥
...
and
λ
k
,
h
→
0
,
k
→
+
∞
, is described above and
(
Z
k
)
are i.i.d. standard normal variables. Moreover, the test I
2
(
h
)
=
(
I
2
n
(
h
))
Carlos Tenreiro
205
deﬁned by the critical regions
{
I
2
n
(
h
)
>
c
α
}
, where
P(
I
∞
>
c
α
)
=
α
, is asymptotically of level
α
and consistent to test H
0
against H
a
.
Remark 1
If the density
f
0
has a compact support
S
and
Q
h
is continuousin
S
×
S
, fromthe Mercer’s expansion for
Q
h
(see Dunford and Schwartz (1963), p. 1088) it followsthat
Q
h
(
u
,
u
)
dP
0
(
u
)
=
∞
k
=
1
λ
k
,
h
and therefore
I
∞
takes the form
I
∞
=
∞
k
=
1
λ
k
,
h
Z
2
k
.
3 Bahadur local efﬁciency
In order to compare the test
I
2
(
h
) with other test procedures, or to compare
I
2
(
h
) testsobtained for di
ﬀ
erent values of
h
, we derive in the following its Bahadur exact slopes
C
I
2
(
h
)
(
f
), for
f
in a neighbourhood of
f
0
. They coincide with the Bahadur approximateslopes (and then with the Bahadur local approximate slopes) derived by Gregory (1980).For the description of Bahadur’s concept of e
ﬃ
ciency, see Bahadur (1967, 1971) orNikitin (1995).Throughout,
·
p
denotes the norm of the Lebesgue space
L
p
(
R
d
,
B
(
R
d
)
,λ
)
=
:
L
p
(
λ
).The proof of the following result is given in Section 5.
Theorem 2
We haveC
I
2
(
h
)
(
f
)
=
b
I
2
(
h
)
(
f
)
λ
1
,
h
(1
+
o
(1))
,
as

f
−
f
0

1
→
0
,
whereb
I
2
(
h
)
(
f
)
=
{
K
h
f
(
x
)
−
K
h
f
0
(
x
)
}
2
dx
,
and
λ
1
,
h
is the largest eigenvalue of the operator A
h
deﬁned by (4).
If
f
0
belongs to a family of probability density functions of the form
{
f
(
·
;
θ
) :
θ
∈
Θ
}
,where
Θ
is a nontrivial closed real interval and
f
0
=
f
(
·
;
θ
0
), for some
θ
0
∈
Θ
, it isnatural to compare a set of competitor tests through its Bahadur local exact slopes when
θ
→
θ
0
.Consider the following assumptions on the previous parametric family:
Assumptions on
{
f
(
·
;
θ
) :
θ
∈
Θ
}
(P)For all
x
∈
R
d
the function
θ
→
f
(
x
;
θ
) is continuously di
ﬀ
erentiable on
Θ
, andthere exists a neighbourhood
V
⊂
Θ
of
θ
0
such that the function
x
→
sup
θ
∈
V
∂
f
∂θ
(
x
;
θ
)
isintegrable on
R
d
.The following result comes easily from Theorem 2, assumption (P) and thedominated convergence theorem.