; 1
2
13
AMS subject classications.
Key words and phrases.
Gaussian time series with long range dependence
Liudas Giraitis , Heidelb erg University
PeterM. Robinson , London Scho ol of Economics
Alexander Samarov ,University of Massachusetts and MIT.
There exist several estimators of the memory parameter in long-memory time
seriesmo delswithmean andthesp ectrumsp eciedonlylo callynearzerofrequency.
In this pap er we give a lower b ound for the rate of convergence of any estimator
of the memory parameter as a function of the degree of lo cal smo othness of the
sp ectraldensityatzero. Thelowerb oundallowsonetoevaluateandcomparedierent
estimators by their asymptotic b ehavior, and to claim the rate optimality for any
estimatorattaining the b ound. A log-p erio dogram regression estimator,analysedby
Robinson (1992), isthen shown to attain the lowerb ound, and is thus rate optimal.
1991 Primary62M15; secondary 62G20, 62G05.
1. Researchsupp orted by DeutscheForschungsgemeinschaft.
2. Researchsupp orted inpart by the ESRC Grant R00023360 9.
3. Partially supp orted by NSFGrant DMS-93{06 245 and by agrantfrom IFSRC at
MIT.
Long range dep endence; semiparametric mo dels; optimal
rates of convergence;lower b ounds.
1
01
0
0
0
0 1
=
1 (1+2 )
n
t
t
t
j
j
r
n
n
r
n
n
=
n
r f g
j j
2 0 2 0
! 2 1 !
2
2 0
2 0
2
!
j j ! 2 1 2
2 0
!1
X
: f
L
; ;; ; ;
L C C ;
X
H =
d = f
;
;
L
;
;
L
=
j=n
: L C O ; ; C ; ; ; ;
n M ; r r
= M
n M
M n =M o
n
range dep endent, stationary,Gaussian timeseries with sp ectral density
(1 1) ( )=
( )
[ ] ( 1 1)
and ( ) , (0 ),as 0. Wewillassume,withoutlossofgenerality,that
the mean of is zero (note that Theorem 2 b elow do es not use this assumption).
The parameter determines the b ehavior of the sp ectrum near zero and is just a
re-expression of the self-similarity parameter = ( +1) 2 and of the fractional
dierencingparameter = 2, see, e.g.,Beran (1994). When = 0, ( ) tends to
anite p ositiveconstantat zerofrequency,while if (0 1) ittendsto innity and
if ( 1 0) it tends to zero.
Severalestimatorsof are now availableb oth for the parametriccase when ( )
is sp ecied for all [ ] up to a nite-dimensional parameter (e.g., Foxand
Taqqu (1986), Dahlhaus (1989) who required (0 1)) and for the semiparametric
case when assumptions are made only ab out the lo cal, for 0, b ehavior of ( )
(as in (1.1) ab ove), which then acts as an innite-dimensional nuisance parameter
(e.g.,Gewekeand Porter-Hudak(1983), K }unsch (1986), Robinson (1992,199 3)).
Perhaps, the b est known estimator of in the semiparametric case is that pro-
p osed by Gewekeand Porter-Hudak(1983) whichisbasedon the linearleast-squares
regression of log-p erio dogram on log (2sin ( 2)), for a certain numb er of Fourier
frequencies =2 close tozero. Robinson(1992) analysedthe asymptoticprop-
ertiesof ageneralizedand mo diedformofthat estimator. It follows fromhisresults
that under certainlo cal smo othness conditions, which includethe assumption
(1 2) ( )= + ( ) as 0 (0 ) (0 2]
the estimatorhas the convergencerate for all ( 1 1),where = ( )=
(2 +1) and arbitrarilyslowly.
In his morerecentpap er, Robinson (1993) analysedanother estimator,originally
prop osed by K }unsch(1987), which was based on the lo cal Whittle approximation of
theGaussianlikeliho o dforthefrequenciesnearzero. Undermuchweakerassumptions
thanthoseforthelog-p erio dogramregressionestimator,heestablishedtheestimator's
asymptotic prop erties, whichimply,under the condition (1.2), the rate with
any suchthat log ( ) = (1).
Note that Robinson (1992, 1993) proves not just the rate of convergence of the
considered estimators but their asymptotic normality with mean zero. By analogy
with many other nonparametric estimation problems, one can exp ect that if the
numb er of Fourier frequencies used is chosen to optimally balance the asymptotic
bias and variance,these estimatorswillattain the rate .
In this pap er weconsider the semiparametricmo del(1.1), (1.2) and givea lower
b ound for the rate of convergence of any estimator of as a function of the degree
(
0 0 n
n
n 0
0
2
0
0
0 2. The lower bound.
0 0
0 0
0 0 0
0
^
( )
0
0 0 0 0
1 (2 +1)
1 r
f
n
f F ;C;K f
r
n
n n
h
n
n
n
=
n h
n
n
n
n n n
f j j
j j j j j j 2 0 g
0 fj j g
1 fj j g
f j 0 j g
2 0
2
2
0
There exists a positive constant such that
where is taken over all estimators of and .
Proof.
n
> F ;C ;K
C K
: F ;C ;K f f C ; <C C ;
< ; K ; ; :
x c
l d> l x d x c x
f
f P
c
: P n f c > ;
r =
f ; ;
C > f F ;C ;K
: f c ; ; ;
:
; < n
c ; ;
h >
: c h :
the rate cannot b e improved on the class of sp ectral densities dened in (2.1).
The lower b ound allows one to evaluate and compare dierent estimators by their
asymptoticb ehavior,and toclaimthe rateoptimalityforanyestimatorattaining the
b ound. We show, in Section 3, that the lower b ound is attained by amo died form
of Geweke-Porter-Hudak's (1983) estimator suggested by Robinson (1992), i.e. that
this estimatoris rate optimalfor the class of sp ectral densitiesdened in(2.1).
For any 0, we dene the class ( ) as in
(1.1) and (1.2), but with the xed b ounds and forthe constants in (1.2):
(2 1) ( )= : ( )= (1+1( )) 0
1 1( ) [ ]
The pro of ofthe following theoremgivingthe lowerb ound forthe rateof conver-
gence of arbitrary estimatorof is clearlyrelated to pap ers of Samarov (1977) and
Hall and Welsh (1984). While the theoremstates the lower b ound for the risk for a
0 1 loss function 1 , it clearly implies similarresult for any loss function
()such that for some 0 ( ) 1 for all .
The notation ( ) in the theorem is used to emphasize that the parameter
corresp onds to thesamesp ectral density whichdenesthe probabilitymeasure .
THEOREM 1.
(2 2) liminf inf sup ^ ( ) 0
inf = (2 +1)
Let ( )=1 [ ],b e the sp ectral density of the white noise. We
assume without the loss of generalitythat 1, so that ( ). Dene
a sequence of\p erturb ed" sp ectral densitiesas follows:
(2 3) ( )= (1+1 ( )) (0 ]
where
(2 4) 1 ( )=
0 0 :=
1
= with some 0,and
(2 5) =1+ log ( )
X
X (
R
0
0
0
2
2
0
0
2
0 0
0
0
0
0 0
0
0 0
0 0
n
n
fn
f
n
n
n
n
n
n n
n n
0 0
0
2 1
1
0
1 2
1
2 2
2
2
2
1 2
( )
0
( )
0
2
0 0
( )
2
2
0
1
0 1
f g f g
f j 0 j g
f g f g 0 f g
f g f g0 0 f g
0 [
f g 0 f g0 0 f g
n n
n
n h
n
n
n
n
f f
n
n
n n
dP
dP
n f n
n
f n n
f f
f
a
f
n
r
n
f F ;C ;K
f n f n f n n
f F ;C;K
f n
a
f n f n n
n n n
r
n n n
f F ;C ;K
f n
a
f n n f n n
For all suciently large
(i) and
(ii) for some constant .
There exist positive constants and such that for all suciently large
(i)
(ii)
f
c ; <
; :
n
f F ;C ;K
f f d Kn K >
P P R n
X ; ;X
f f
K K n
m E K <
E m K < :
P P
A a>
: P A e P A
M
a
;
M K K
U f n f c
P U f P U f P U f :
P U f e P U f
M
a
P U f :
f f f h n U f U f
c<=
P U f e P U f
M
a
P U f :
( )=
0
1
LEMMA 1. (cf. Hall and Welsh(1984))
( )
( ( ) ( )) 0
The pro of of the lemmais giveninSection4.
Denoteby and theprobabilitymeasureson generatedby observations
=( ... ) of the Gaussian stationary sequence with mean zero and sp ectral
densities and resp ectively,and let 3 =log ( ( ))denote the log likeliho o d
ratio.
LEMMA 2. (cf. Lemma1 inSamarov (1977))
:= 3 ;
:= (3 )
This lemma, the pro of of which is also given in Section 4, guarantees that the
measures and are close in a certain sense. It is easy to check, for example,
that for any event and any 0
(2 6) +
with = + .
Denote now the event
( )= ^ ( )
and observe that for any , 0 1,
sup ( ) ( ) +(1 ) ( )
Applyinghere (2.6), we get
sup ( ) ( ( ) )+(1 ) ( )
Now,since is chosen such that ( ) ( )= = , ( ) ( )=,
the certain event,for any 2, and wehave
sup ( ) (1 ( ) )+(1 ) ( )
2
R
0 0
f g 0
2 0
0 2
2
0
0
0
0 0
0
( )
2
1 2
0 0
0 0
0 0
1 2
2
f F ;C;K
f n
a
a
=
n
r =
r Remark2.1.
Remark2.2.
Remark 2.3.
Remark 2.4.
3. Upper bound.
= e
P U f
e
e
M
a
;
a>M
f
f d
=
F ;C ;K ;
; ;
C
C
F ;C ;K
F ;C ;K
n n
f
L
n
Cho osing =1 (1+ ), we get
sup ( )
1+
(1 )
and the conclusion of the theorem follows if we cho ose .
Whiletheconstructionofthe p erturbation ( )in(2.3)issimilar
to that of Hall and Welsh (1984), it is a bit simpler here since, unlike Hall and
Welsh (1984) who work with probability densities, we allow small p erturbations in
( ) . This last dierence is also apparently related to the fact that their rate
is determinedby the ratio while inour case it dep ends only on .
Since the rate in (2.2) is indep endent of , it sues for the pro of to consider
p erturbations of a \base" sp ectral density in ( ) with any ( 1 1); we
havechosenthe\base" densitywith =0sincethischoicesimpliesthepro of. Note
that the lower b ound would remain valid for classes of densities with the range of
values of smaller than ( 1 1), e.g. for [0 1).
Note also that the assumedGaussianity of the pro cess isnot a restriction for the
resultofTheorem1,sincethelowerb oundestablishedundertheGaussianassumption
willautomatically hold for broader classes of pro cesses.
Eventhough weconcentratehere on theestimationof thememory
parameter , in practice one also has to estimate the scale parameter in (2.1).
Robinson (1992, 1993) considered several estimates of and analysed their asymp-
totic prop erties. A lower b ound, similar to (2.2) but with an additional logarithmic
term,forthe rate of any such estimatecan b e easilyobtained by onlyslightly mo di-
fying the argument givenhere.
The parameter dening the class ( ) determines the
lo caldegreeof smo othness of the sp ectraldensities inthe class. Notethatif grows
to innity,the class ( ) b ecomecloser and closer to the purelyparametric
family,and rate approaches the parametricrate .
The result of the theorem can b e also formulated in terms of
mo dulus of continuityof ( ) dened similarlyto Donoho and Liu (1991), but with
the norm whichapp ears to b e morenatural here than the Hellinger norm.
In this section we showthat the rate , givenin the lower
b ound in Theorem 1, is attainable and thus optimal. We b elieve that it can b e
attained by a numb er of estimates of including those discussed in Section 1. We
consider here the mo died version of Geweke and Porter-Hudak's (1983) estimator
which Robinson (1992) showed to b e asymptotically normal. It app ears to aord
the simplest pro of in the present circumstances; a slightly more general pro of will
n
0
0 0
Proof.
X
P P P
P
P
P
P
P
X
X 0
0 0
0
0
2
2
0
0
0
0
0
0
j j
0 0
0
f g
!1
0 1
f g f g
0
j j
2
!1
2 2 !1
1 2
=1
2
1
=+1
2
0
0
3
0
0 2
3
2
0
0 2
0
( )
( )
2 2
1 2
2
2
0
2
2
2 2
0 0 0
= n
k k
ik
j
j
k k
m
k l
lmn
j
j j
j j
n
r r
r
n
j
n
l;m J r;D
f F ;C;K r
f lmn
=
j j j
T
j j
lmn
j j j
j j
j j j
n
j j
j j j
r
n w
n
X e ; I w :
l m l < m n= j=n
j m l k f
:
I
:
D > r
J r;D l ;m l <m n=
n
D
l
D n
n n
D
m D n
n m
J r;D l
I j n
n E f < :
v w = C v R e v ;Im v
f
u
;
u v : ::: u
l ;m J r;D
: m O
l n
m
m;
n
E u O n
l ;m J r;D f F ;C ;K n
smallerasymptoticvariances), whilearatherdierentandmoredicultpro of would
b e neededfor the estimates considered by K }unsch (1987), Robinson (1993).
Dene the discreteFourier transform and p erio dogram
( )= 1
(2 )
( )= ( )
Let and b e integers such that 1 2, and put = 2 ,
=log ( ) log , where = . We estimate = ( ) by
(3 1) ^ =
log ( )
For any 1 and as denedin Theorem1,the set
( )= : 1 2;
log ( )
log ( )
;
is non-empty for suciently large. In (3.1), is a bandwidth numb er which
achieves its \optimal rate" in ( ), while is a limiting numb er designed to
avoid the anomalousb ehaviour of ( )for nite as , seeK }unsch(1986).
THEOREM 2.
liminf max sup (^ ( ))
Put ( ) = ( ) ( ) and =( ( ) ( ) ) . Because
=0,
^ = ( )
where =log ( ) + ,where =05772 isEuler'sconstant,so thatthe have
approximatemeanzero,see(3.10)b elow. FromRobinson (1992), for ( )
(3 2) = (1+ (
log ( )
))
as . Thus it sues to showthat
( ) = ( )
uniformlyover ( ) and ( )as . (In the rest of the
pro of, we willuse the word \uniformly"in the samesense as here without rep eating
0
0
0
0 X
X
Z
Z
" #
0 0 0
2 2 2
:
2
0
0 0
2 2
1 2
2
1 2
2 1 2 2
2 1 2
2
2 2
0
1
1 12
21 2
!1
2
2 !1
j j j 0 j j j
2
j j j j
2 0 2
2 1
jj jj jj jj
j j
j j
!1
0 jj jj 1
2
2 n
j j
j
r
k ;j l<k <j m
j k j k
r
j j
n
f j f j
f j k f j k
lmn d
d
T
E
E
T =
j j
=
=
j
j j j
j
E
n
jk
T
j T
k T
jk n
: E u O n
: E u u O n :
v Ev j <n
l ;m J r;D
f F ;C ;K n
: E v E v O
j
j
j
n
j l ;m
: E v v E v v O
j
k
j
n
k l ;j ; j l ;m
I
d d d
x x ;x g x x A
Tr A A A
: Eu g x x dx;
v
I = o n n
g x x dx<
m J r;D
j > k v ;v
j k
: ,
(3 3) ( ) = ( )
and
(3 4) ( )= ( )
By Gaussianity, the moments in (3.3) and (3.4) can b e analyzed in terms of the
rsttwomomentsofthe . Wehave =0,1 andthefollowingprop erties
essentiallyfollowfromTheorem1ofRobinson(1992): uniformlyover ( )
and ( )as
(3 5) ( ) 1 + ( ) = (
log
+( ) )
uniformlyin [ +1 ] and
(3 6) ( ) ( ) + ( )( ) = (
log
+( ) )
uniformlyin [ +1 1] [ +2 ].
The pro of of (3.3) and (3.4) uses techniques similar to those in the metho d-of-
moments pro of of asymptotic normality of ^ in Robinson (1992). Denote by
the identity matrix, by () the -variate standard normal density, and for
a 2-dimensional vector = ( ) put ( ) = log ( )+ , where =
( ( )) denotes the Euclideannorm of a matrix .
To prove(3.3), write
(3 7) = 6 ( ) (6 )
where 6 is the covariance matrixof and 6 is its determinant. It follows from
(3.5)that 6 = 2+ (1) uniformly,as ,so that for sucientlylarge(3.7)
is b oundeduniformlyby
3
2
( )exp(
1
2 )
from the prop erties of the normal distribution. Then (3.3) follows from (3.2) and
( ).
To prove (3.4), take and denote by 6 the covariancematrixof ( )
and, suppressingreferenceto and ,8=6 ,partitionedinto 2 2submatricesas
8=
8 8
8 8
0 0
" # " #
Z
Z
Z
Z
Z Z
X X X
T T T
j k
= =
= = T
=
i
=
i
T T
E
T
E
= T
=
i
E
E
n
k ;j l<k <j m j k
j j
j j 1
2
12
21
1 2
4 1 2
1 2
4 1 2
1 2 2
=1
2 1 2
2 2
2
4 1 2
4
2 2
2 1 2
2
2
2
2
2
2 2
0
:
2 2
3
2 2
j j
j j 0 0
j j
j 0 0 0 j jj jj
jj jj
!1
!1
j j
!1
p
jj jj
!1
p
j jjj jj p
1
! 1
2
j j j j
; :
z x ;y
E u u g x g y z dz
: g x g y z z z dz
: g x x dx :
>
z z z z O
j
k
j
n
z ;
z z z
n
g x g y z z zdz
n I o n
O O j =k j=n
n i ;
x x O
j
k
j
n x
n
: g x x dx ; g x x x dx;<
O j =k j=n n
l ;m J r;D
j
k
j
n
m
l
m
n
m
~
8=
8 0
0 8
8=
0 8
8 0
With =( ) ,
( )= 8 ( ) ( ) (8 )
(3 8) = 8 ( ) ( ) (
~
8 )[exp(
1
2
8 ) 1]
(3 9) +8 5 [ ( ) (8 ) ]
Using(3.5)and(3.6)and arguingasinthepro ofofTheorems2and3ofRobinson
(1992), wehave for some 0
exp(
1
2
8 ) 1 1
2
8 = ( [ log( )
+( ) ] exp( ))
~
8 4
uniformly, as , while also
( ) ( ) (
~
8 )
8 =0
for all large enough, b ecause (3.5) and (3.6) imply that
~
8=2 + (1) as .
Likewise 8 = (1)andso itisreadilydeducedthat(3.8)is ( (log ( ) ) +( ) )
uniformly, as . To estimate(3.9), note from(3.5) that, for =1 2,
(8 )= ( 2 ) 1+ ( ( log ( )
+( ) ) )
uniformly, as (cf. (5.26) ofRobinson (1992)).
Because
(3 10) ( ) ( 2 ) =0 ( ) ( 2 )
it follows that (3.9) is uniformly ( (log ( ) ) + ( ) ) as . Then, for
( )
[(
log( )
) +( ) ]
2(log )
+( )
2
n X
0
0
0 0
0 Proof of Lemma 1.
Remark 3.1.
4. Proof of lemmas.
2 1 2
2 2
2
0 0
0
1 2
2
2 2
2
1
2
0
2
j j j j
1
2
j j j j
0
0 0
0 0
j j
j j
j j
=
r
j j
lmn
n
n
n
n n n n
h
n n
n
n n n n n n n
n
n
n n
n
n n n
=
n
n
n n
n
n
O
m
l
m
m
n
O
m n
l
m
n
O n
f
;
F f = h
h
h
h =
F ;C ;K
=
= ; >
: K ; :
c h O h ;
: h O h ;
: h O h h O h
h
O h :
= = = e h
K
h
e
K h K K ;
= (
(log )
[ ] + )= (
(log )
+ )= ( )
to verify(3.4).
Note that (3.5) and (3.6) can b e deduced via a somewhat sim-
pler pro of than those in Robinson (1992), where that are not b ounded outside a
neighb orho o d of zero are p ermitted. In Theorem 1 of Robinson (1992), (0 2] is
assumed. Observethatifthe class includes ( )ofthe form(2sin ( 2)) ( )for
a b ounded function ( ), as in case of fractionally integrated autoregressive moving
averagepro cesses, then 2no matterhowsmo oth ( ) is,for example,evenwhen
( ) 1. On the other hand, it seemsthat if we replace by 2sin ( 2) in
the denition of ( ) then = is p ermitted in the latter situation, and
to take advantage of this we would also need to replace log ( ) by log (2sin 2) in
the denition of ^ , as inGeweke-Porter-Hudak's (1983) original form. The p oint
is that and 2sin 2 are interchangeablewhen (0 2], but not when 2.
To prove the claim(i),it is enough to show that in (2.3) and
(2.4)
(4 1) 1 ( ) for
Combiningthe estimate
=1 log ( )+ (( log ( )) )
whichfollows from(2.5), with the estimate
(4 2) 1 log ( )= (( log ( )) )
whichholds uniformlyfor , wehave
(4 3) 1 ( )=(1 log ( )+ (( log ( )) ))(1+ log ( )+ (( log ( )) )) 1
= log ( )+ (( log ( )) )
Now, as in Hall and Welsh (1984), it is sucient to notice that the maximum of
( ) log( ) for isachievedat = . Then,since = ,
(4.3) impliesthat forsome ( ):
1 ( )
1
+ ( )( log ( )) ( )
2
X X
X
Z Z Z
Z Z
Z Z
Z
ProofofLemma2 0
0
0 0
0
0
0
3
0 3
3
0 3
jj jj
n n
n
k
n
n
n
k
n
n
n
k
n
n
n
k
n
n
n
E 0
0 2
0
0 2
0
2
0
2 2
1 2
1
0 1 1 0
2
2 2 2
2
0
2 4 4 2 +1
0
1
1
1
1 2 1
=1
0 0 0
0 0
0
2
0 j j 0 0 0
j j
0
0 j j
0
j j
0
jj jj jj jj
jj jj jj jj
n
n
n h
n h
n
h
n
h
n
n
n n
n
n
n
n
n
n
n n
n n n
n
n n
T
n
n
T
n n
n
n n n
n f n n n
n n n n n
n
n
n
n n
n
n
n n
sp
x
E
E E
f f d f f d c d
c d c d I I ; ;
k C
n I C =n n n
I c d h
O h d
h
d O h K K =n:
A B n n
f f A I
B I B ;
X ; ;X n ;:::; B
B
D B I
m E Tr D B :
A I D A A
d
d
A Tr A
d
d
A ;
: m Tr D A D
Tr D A :
A Ax A
: i Tr CD C D :
To provethe second claim,we write
( ( ) ( )) =2 ( ( ) ( )) =2 ( 1)
=2( ( 1) + ( 1) )=:2( + ) say
for some 2(2 +1). It is easy to check that for suciently large and some
( ), for all ( ). Arguing similarly to the pro of of claim (i), we
have,using (2.5) and (4.2),
= ( 1) = ( log ( )+ (( log ( )) ))
2 log ( ) + ( log ( )) =
closelyfollowsthepro ofofLemma1ofSamarov(1977). Denote
by and the covariance matrices corresp onding to the sp ectral densities
and resp ectively. Of course, = , the identity matrix. The log likeliho o d
ratio 3 has the form
3 = 1
2
(log +( ) ( )( ))
where =( ... ), isthe -vector of means( ),and denotes the
determinantof the matrix .
Denoting also = , wehave
= 3 =
1
2
( ( ) log )
Set ( ) = + , 0 1. Applyingmean value theorem to (0) (1)
and using the fact that
log ( ) = ( ( ) ( ))
see,e.g. Davies (1973), weobtain, for some0 1,
(4 4) =
1
2
( ( ) )=
2
( ( ))
Denote by = sup the sp ectral norm of a matrix . The
following four inequalitiesare well known, see,e.g. Davies (1973):
(4 5) ( ) ( )
n
2
P P
Z
Z
Acknowledgement.
1
= =
2
2 2
1
[ ]
2 1
2
0 2
1
2 2 1 2 2
2
0
1
01
0
1
01
0
0
20
0 3
0
3 3
3 0 3
0
E E sp
j k j;k n
j j
ij
j j
j j
E
sp
;
n n
E n
sp
n
E
n
n n
n
sp
n
f n n n n
n E
n n
E
f g 2
1
jj jj
jj jj j j
jj jj jj jj
jj jj 0
!1
0
2 0 jj jj
0 jj 0 jj jj 0 jj
!1
C c n n
c e c c c <
: iii C
n
d
C
: iv C = :
: m D A :
D
n
f f d K;
n
f A
; A
E m B I B B I K
n
Let = b e a To eplitzmatrixgenerated by afunction ( )=
, = ,and . Then
(4 7) ( )
2
( )
and, if is p ositivedenite,
(4 8) ( ) sup 1 ( )
Applying(4.5) and (4.6)in (4.4), we get
(4 9)
1
2
( )
But by (4.7) and Lemma1 (ii)
2
( ( ) ( ))
~
as .
Itiseasyto seethatthefunction1+ ( ( ) 1)generatingthematrix ( )is
b oundedawayfrom0for [ ]and0 1. Therefore,by(4.8) ( )
is b ounded,whichtogether with (4.9) gives the rst claimof Lemma2.
To prove the second claim,we use the well-known expression for the variance of
the Gaussian log likeliho o d (see,e.g.,Davies(1973)), (4.7), and Lemma1:
= (3 ) =
1
2
( ) =
1
2
as .
A part of this pap er was written while the rst and third
author (LG andAS) werevisitingthe Institut f }urAngewandte Mathematik,Univer-
sity ofHeidelb erg. Theywouldliketo thank theinstituteandin particularProfessor
Rainer Dahlhaus for providingthe ideal working environment.
5
17
19
14
4
12
23
4
13
Statistics for long-memory processes
Adv. in
Appl. Probab.
Ann.
Statist.
Ann. Statist.
Ann. Statist.
J. Time Ser. Anal.
Ann. Statist.
J. Appl. Probab.
Proc. 1st World
Congress of the Bernoulli Society
Problems
of Info. Transm.
Beran,J. (1994). . Chapman &Hall,New York.
Davies, R.(1973). Asymptoticinference instationary Gaussian time-series.
469-497 .
Dahlhaus, R. (1989). Ecientparameter estimationfor self-similar pro cesses.
1749-176 6.
Donoho, D. and Liu,R. (1991). Geometrizing rates of convergence,I I.
633-667 .
Fox , R. and Taqqu, M. (1986) Large-sample prop erties of parameter estimates for
strongly dep endent stationary Gaussian timeseries. 517-532 .
Geweke, J. and Porter-Hudak, S. (1983). The estimation and application of long
memorytimeseries mo dels. 221-238 .
Hall, P. and Welsh, A. (1984). Best attainable rates of convergence for estimates of
parameters of regularvariation. 1079-1 084 .
K }unsch, H.(1986). Discriminationb etweenmonotonic trendsand long-range dep en-
dence. 1025-1 030 .
K }unsch, H. (1987). Statistical prop erties of self-similar pro cesses.
67-74, VNU SciencePress.
Robinson, P. (1992). Log-p erio dog ram regression of time series with long range de-
p endence. Preprint.
Robinson, P.(1993). Gaussian semiparametricestimationof long range dep endence.
Preprint.
Samarov,A.(1977). Lowerb oundfortheriskofsp ectraldensityestimates.
67-72.
