Table Of ContentAsymptotic near-efficiency of a “Gibbs-energy”
estimating function approach for fitting Mat´ern
covariance models to a dense (noisy) series.
D.A. Girard
9
0
0 CNRS and University J. Fourier, Grenoble
2
p
e
S Abstract
5
Let us call “Gibbs energy” the quadratic form occurring in the maxi-
] mum likelihood (ML) criterion when fitting a zero-mean multidimensional
T
Gaussian distribution to one realization. We consider a continuous-time
S
. Gaussian process Z which belongs to the Mat´ern family with known regu-
h
t larity index ν 1/2. For estimating the range and the variance of Z from
a ≥
noisy observations on a dense regular grid, we propose two simple estimat-
m
ing functions based on the conditional Gibbs energy mean (CGEM) and the
[
empirical variance (EV). We show that the ratio of the large sample mean
1
squareerror of the CGEM-EV estimate of the range-parameter over the one
v
6 of its ML estimate, and the analog ratio for the variance-parameter, both
4 converge (when thegrid-step tends to 0) toward aconstant, only function of
0
ν, surprisingly close to 1 provided ν is not too large. Extensions of this ap-
1
. proach, which may enjoy a very easy numerical implementation, are briefly
9
0 discussed.
9
0
: 1. Introduction
v
i
X
We consider time-series of length n obtained by observing on a dense regular
r
a grid, a continuous-time process Z which is Gaussian, has mean zero and an au-
tocorrelation function which belongs to the Mat´ern family with regularity index
ν 1/2. This family is commonly used, e.g.in geostatistics (see Stein 1999); recall
≥
that ν = 1/2 correspond to the well known exponential autocorrelation (in other
words, Z is a stationary Ornstein-Uhlenbeck (O.U.) process). The general defini-
tionof Mat´ern processes can beeasily formulated in terms of theFourier transform
of their autocorrelation function, namely the spectral density over ( ,+ ):
−∞ ∞
1
c θ2ν
ν
f (ω) = bg (ω), where g (ω) = . (1.1)
ν∗,b,θ ν∗,θ ν∗,θ (θ2 +ω2)21+ν
Γ(ν+1)
In this paper the constant cν = √πΓ(ν2) is chosen so that ∞ gν∗,θ(ω)dω = 1. Thus
−∞
b is the variance of Z(t) and θ is the so-called “inverse-range parameter” (in fact,
R
it is 2ν1/2/θ which can be interpreted as the effective range or “correlation length”,
cf Stein (1999, Section 2.10); we will often drop the term “inverse”.
We are concerned here with dense grid in the sense that the interval δ between
two succesive observations is small relatively to 1/θ. The considered processes
being mean square continuous, this means that two successive ideal (i.e. with-
out measurement error) observations are “strongly” correlated. See Stein (1999,
Chapter 3), for such a setting which shows that standard large-n asymptotic anal-
ysis followed by a (less standard) small-δ analysis yields useful insights and good
approximations for various real (finite-size) problems.
In this article, we assume that ν is known and we mainly study settings where
there are Gaussian i.i.d. measurement errors, or, equivalently for the paramet-
ric inference point of view we take here, there is a so-called nugget effect. No-
tice that introducing such a nugget effect is actually not too restrictive since, of
course, data always have a finite number of digits, so one may easily accept to
include, in the parametric model, a known nugget equal to the squared finite pre-
cision; furthermore this is known to often remedy to numerical instabilities due
to ill-conditionning of the linear systems we may encounter in the no-nugget case.
However, in this first study, we restrict ourselves to the case where the suspected
measurement errors have a known variance. We thus assume that (possibly after
appropriate rescaling) the observations is a vector y of size n whose conditional
law given Z is
y z N(z,I) where z N(0,b R ),
| ∼ ∼ 0 θ0
with I denoting the identity matrix and R the autocorrelation matrix of the col-
θ
umnvectorz = (Z(δ),Z(2δ),...,Z(nδ))T,i.e.theToeplitzmatrixwithcoefficients
given by
[R ] = K (δ j k ), j,k = 1, ,n, where K (t) = ∞ g (ω)eiωtdω.
θ j,k ν,θ | − | ··· ν,θ ν∗,θ
Z
−∞
Recallthatexpressions intermsofBesselfunctionsarewell knownforK (seee.g.
ν,θ
Rasmussen and Williams (2006) for explicit very simple expressions for ν = 3/2
and 5/2).
It is often of great interest to be able to “effectively reduce” the number of
parameters, especially when computing the likelihood function is costly. This is
classically done in the no-nugget case (i.e. bR +I is replaced by bR , and y by z,
θ θ
2
in the likelihood function) by noticing that the maximizer b of the likelihood for
any fixed θ is simply:
ˆb(θ) = (1/n)zTR 1z. (1.2)
θ−
See Zhang (2004) also for numerical improvements produced by such a “concen-
tration” of the likelihood. Zhang and Zimmerman (2007) recently proposed to use
the classical weigthed least square method (not statistically efficient but much less
costly than maximum likelihood (ML)) to estimate the range parameters, next, to
plug-in these parameters (the θ here) in the likelihood which is then maximized
ˆ
with respect to b (the solution, say b (θ), being the explicit (1.2) in the no-nugget
ML
case, obtained iteratively by Fisher scoring otherwise). The idea underlying this
method is that, at least for the infill asymptotics context, even if θ is fixed at a
wrong value θ , the product ˆb (θ )θ2ν still remains an efficient estimator of b θ2ν
1 ML 1 1 0 0
(see Du et al. (2009) for recent results of this type).
The method we propose here, firstly reverses the roles of variance and range,
in that it is based on a very simple estimate for the variance, namely the empirical
variance in the no-nugget case, and its corrected version for biais otherwise, which
is simply defined by ˆb := n 1yTy 1. Secondly we propose to replace the
EV −
−
maximization of the likelihood w.r.t. θ by the very simple following estimating
ˆ
equation in θ, in the with-nugget case: solve, with b fixed at b
EV
yTAb,θ(I Ab,θ)y = trAb,θ where Ab,θ = bRθ(I +bRθ)−1. (1.3)
−
In the no-nugget case, this equation in θ is simply replaced by zTR 1z = nˆb .
θ− EV
Onemay call“Gibbs energy” (GE) ofthe underlying process the quantity zTR 1z
θ−
and it is easily seen that b yTA (I A )y+tr(I A ) is the conditional
b,θ b,θ b,θ
− −
Gibbs energy mean (CGEM) obtained by taking the expectation of zTR 1z, con-
θ−
(cid:0) (cid:1)
ditional on y, for the candidate parameters b,θ. So equation (1.3) in θ will be
called the CGEM-EV estimating equation (GE-EV in the no-nugget case) and we
ˆ
will denote by θ this new range parameter estimate.
GEEV
Notice that it is easily checked that equation (1.3) is equivalent to equate to 0
the derivative w.r.t. b (and not w.r.t. θ!) of the likelihood. This proposal is based
on the following two ideas. First, since it is quite plausible that the above idea,
underlying Zang and Zimmerman (2007)’s proposal, remains true for a random
ˆ
θ, then, instead of “fixing” θ at θ , one may as well adjust θ so that b (θ)
1 ML
ˆ
coincides with a given value b for the variance; and, denoting θ the so-obtained
1 1
θ (thus (b ,θˆ ) must cancel the b-derivative of the likelihood), the product b θˆ2ν
1 1 1 1
will plausibly be an efficient estimator of b θ2ν. Second, it is known that, at
0 0
least in the no-nugget case, the moment Z(δj)2 yields a successfull estimating-
equation in the case ν = 1/2 to estimate θ when b θ2ν is known (see Kessler
Pˆ 0 0 0
(2000)). So it is tempting to select b := b , the emprical variance estimate of
1 EV
b . The second point is an admittedly weak justification of this method. However,
0
3
notice that Yadrenko (Chapter 4 Section 3) has studied, for quite general, mean
squarecontinous, stationnaryisotropicrandomfieldsthepropertiesofacontinuous
version of the empirical variance in the periodic case, i.e. the Lebesgue integral
over sphere of Z(t)2, and he has listed appealing properties of such estimates of
the variance of the field. The theorical results we give in this article, will provide
in our context, a much stronger, and rather unexpected, justification, for ν not too
large (which is the case in numerous applications, see Stein (1999), Rasmussen et
al. (2006)).
Of course, in our time-series setting, there now exist rather good implemen-
tations of ML; see Chen et al. (2006) for the case when the correlation is rather
strong. However, our objective here is to provide first insights into the capability
of this method for more computationally complex settings. Indeed, this approach
is not limited to observations on one dimensional lattice, and is potentialy not lim-
ited to regular grid (a weighted version, with Rieman-sum type coefficients, of the
empirical variance shoud then be used instead) and to homogenous measurement
errors: some successfull applications for two-dimensional Mat´ern random fields,
using a randomized-trace approximation to trA , are described in Girard (2009),
b,θ
along with some theoretical properties. This approach might be, in principle,
applied to other two-parameters models than the Mat´ern family (of course it is
presently restricted to scalar θ). Encouraging experimental results are obtained in
Girard (2009) for the commonly used spherical autocovariance functions.
The rest of this article is structured as follows. Section 2 lists additional no-
tations. In the asymptotic framework we adopt here (which may be thought as
intermediate between the infill and increasing-domain frameworks) we first show,
in Section 3, that, even when b is arbitrarily fixed to b , the CGEM estimating
1
equation is a quite well-behaved estimating equation: it converges toward a mono-
tonic equation whose the rootθ satisfies b θ2ν = b θ2ν. Next we study in Section 4
1 1 1 0 0
ˆ
what is sacrificed when plugging-in the simple b in the CGEM estimating equa-
EV
tion, as compared to ML. The performance-loss is classically quantified by the
asymptotic mean square errors of the two components of the CGEM-EV estimate
as related to those of the ML estimate. We show that what is sacrificed, which
depends on ν, is quite reasonable provided ν is not too large, independently of b
0
and θ . Indeed asymptotic efficiency is reached as ν decreases to 1/2. Proofs are
0
outlined Section 5.
2. Further notations
Of course “time” could be replaced everywhere by “space”: we choose this
vocabulary since we use in several places of the paper the classical time-series
theory. Thus we assume everywhere without loss of generality that Z defined by
δ
Z (i) = Z(δi) is observed at times i = 1,2, ,n. The spectral density on ( π,π]
δ
··· −
4
of Z is thus fδ = bgδ with
δ ν,b,θ ν,θ
1 ∞ +2kπ ∞
gδ ( ) := g · = g ( +2kπ) where α = δθ,
ν,θ · δ ν∗,θ δ ν∗,α ·
k= (cid:18) (cid:19) k=
X−∞ X−∞
the equality between the two sums resulting from the particular ”variance-scale”
parameterisation of the Mat´ern family. We will also denote simply by g ( )
ν,α
·
the second sum (α is the range-parameter for the series on Z). Simple closed
expressions for gδ are available only when ν 1/2 is a small integer, says 0,1 or
ν,θ −
2 : they then coincide with particular constrained ARMA spectral densities.
Let us define the “Bayesian filter” aδ ( ) and its “un-aliased” version a ( ),
b,θ · ∗b,α ·
for α = δθ, by
bgδ ( ) bg ( )
aδ ( ) := ν,θ · , a ( ) := ν∗,α · .
b,θ · bgδ ( )+(2π) 1 ∗b,α · bg ( )+(2π) 1
ν,θ · − ν∗,α · −
As is well known aδ is the spectral characteristic of the “n = + ” version of
A (a convolutionb,θof series defined on Z) so that, e.g., lim ∞n 1trA =
b,θ n + − b,θ
(2π) 1 π aδ (λ)dλ for any fixed δ,b,θ. → ∞
− π b,θ
A fu−nction which will play an important role in this article is the derivative of
R
log(gδ ( )) w.r.t. θ, and its un-aliased approximation (up to a factor δ):
ν,θ ·
hδ := ∂log(gδ )/∂θ, h := ∂log(g )/∂α.
ν,θ ν,θ ∗ν,α ν∗,α
For any f : [ π,π] R, s.t. π [aδ (λ)]2f(λ)dλ = 0, we define the weighted
− → π b,θ 6
coefficient of variation of f by −
R
2
1 w f 1 wf 1 wf2
J (f) := R w − R w = R w 1, where w := (aδ )2.
δ,b,θ R h 1 (cid:16)wf R2 (cid:17)i 1 R wf 2 − b,θ
R w R w
(cid:16) (cid:17) (cid:16) (cid:17)
R R
Above and throughout this paper, “ ” will denote integrals over [ π,π]. We will
−
also use the notation g (resp. h ) for the function gδ (resp. hδ ) when θ = θ
0 0 R ν,θ ν,θ 0
and also a := aδ , and J ( ) := J ( ).
0 b0,θ0 0 · δ,b0,θ0 ·
3. Some asymptotic properties of the CGEM estimating equation
In this section, we established some consistency and identifiability properties
enjoyed by the CGEM estimating equation in θ. They are merely encouraging pre-
liminary results. We expect to give more complete results elsewhere. To simplify
5
our statement, we consider, the normalized version yTA (I A )y/trA = 1
b,θ b,θ b,θ
−
which is, of course, an equivalent estimating-equation.
For any positive δ,b,θ,b ,θ , we can define the following weighted mean:
0 0
1 π b gδ (λ)
ψ(δ,b,θ,b ,θ ) := [aδ (λ)]2 0 ν,θ0 1 dλ.
0 0 π aδ (λ)dλ b,θ bgδ (λ) −
π b,θ Z π ν,θ !
− −
R
Theorem 3.1. For any fixed b,θ, we have the following convergence in probability:
yTA (I A )y
b,θ b,θ
− 1 ψ(δ,b,θ,b ,θ ) as n ,
0 0
trA − → → ∞
b,θ
and ψ has a very simple small-δ equivalent:
b θ 2ν
ψ(δ,b,θ,b ,θ ) 2ν(2ν +1) 1 0 0 1 as δ 0.
0 0 −
→ bθ2ν − →
(cid:18) (cid:19)
The first part of this theorem is in fact not restricted to the Mat´ern fam-
ily; this is a rather direct consequence of universal results of time-series the-
ory about quadratic forms constructed from (product of power, possibly nega-
tive, of) Toeplitz matrices. In order to be allowed to apply such classical results
(for instance, those stated by Azencott and Dacunha-Castelle (1986) and used in
their analysis of maximum likelihood estimation, or its Whittle approximation),
it is sufficient that (θ,λ) gδ (λ) be three-times continuously differentiable on
7→ ν,θ
Θ [ π,π]where Θ isa compact interval not containing 0; andthis canbechecked
× −
by applying the classical Weierstrass M-test.
The secondpartuses several appromixationstointegralswhich canbeobtained
by the technics of Section 5.
Theoreme 2.1 thus says that the limit of the normalized CGEM equation has
a small-δ equivalent which is a very simple monotonic function of bθ2ν. Clearly if
b is fixed at any value b , then the unique root θ of this large-n-small-δ equivalent
1 1
equation will satisfy b θ2ν = b θ2ν. This gives support to the first of the two
1 1 0 0
underlying ideas given in the introduction.
4. Mean square error inefficiencies of CGEM-EV to ML for the variance
and range parameters
ˆ ˆ
Let (b ,θ ) be a maximizer of the likelihood function on B Θ where B
ML ML
×
and Θ are compact intervals not containing 0 and such that (b ,θ ) is in the
0 0
6
interior of B Θ. Then, since the classical identifiability and regularity conditions
×
are well fulfilled, for any fixed δ > 0, it is a well known result of times-series
ˆ ˆ
theory (Azencott and Dacunha-Castelle (1986)) that (b ,θ ) is a.s. consistent
ML ML
and satisfy:
ˆb b 0 σ2 σ
n1/2 ML 0 N ,4π 1 12
(cid:18)(cid:20) θˆML (cid:21)−(cid:20) θ0 (cid:21)(cid:19) →D (cid:18)(cid:20) 0 (cid:21) (cid:20) σ12 σ22 (cid:21)(cid:19)
where
σ2 b2 a2h2
1 0 0 0
σ12 = ∫ a20h0 −2J0(h0)−1 −b0R a20h0 .
σ2 a2
2 (cid:0) (cid:1) R 0
Note that the factor (∫ a20h0)−2J0(h0)−1 is the deRterminant (> 0 since h0(·)
cannot be a constant function) of the information matrix and has this form only if
a2h = 0. And the same regularity conditions on our time-series Mat´ern model
0 0 6
fδ are also sufficient to show (by the usual Taylor series argument) the first part
Rν,b,θ
of the following theorem:
Theorem 4.1. Let (ˆb ,θˆ ) be a consistent root of the CGEM-EV estimating
EV GEEV
equation (1.3). If a2h = 0 then
0 0 6
R ˆ
b b 0 v v
n1/2 EV 0 N ,4π 1 12
(cid:18)(cid:20) θˆGEEV (cid:21)−(cid:20) θ0 (cid:21)(cid:19) →D (cid:18)(cid:20) 0 (cid:21) (cid:20) v12 v2 (cid:21)(cid:19)
where
v = b2 a 2g2
1 0 −0 0
v = b c a2h −1 , and c = J (g /a2).
12 −R0 0 0 0 0 0 0 0
v2 = c0 a20(cid:0)R a20h0(cid:1)−2
When δ → 0 we have a20hR0/ (cid:0)Ra20 →(cid:1)2ν/θ0; and defining by Iδ1,b0,θ0 := v1/σ12
(resp. I2 := v /σ2) the asymptotic mean square error inefficiency of CGEM-
δ,b0,θ0 2 2 R R
EV to ML for b (resp. for θ ), these 2 inefficiencies have the following common
0 0
limit
√π 2ν +1 2 Γ ν + 1 2Γ 2ν + 1
Ii 2 2 as δ 0, for i 1,2 .
δ,b0,θ0 → 2 2ν Γ(ν)2Γ(2ν +1) → ∈ { }
(cid:18) (cid:19) (cid:0) (cid:1) (cid:0) (cid:1)
The first rather surprising fact is that this large-n-small-δ inefficiency is not
function of the underlying b , nor of θ . Secondly, Table 4.1 clearly demonstrates
0 0
that the CGEM-EV estimates are asymptotically nearly efficient provided ν is not
too large. Asymptotic full-efficiency is reached for ν close to 1/2.
7
ν 1/2 1 3/2 2 5/2 3 7/2 4
lim Ii 1 1.04093 10 1.18596 63 1.33174 1716 1.46727
δ→0 δ,b0,θ0 9 50 1225
Table 4.1. Limit of asymptotic MSE inefficiencies Ii ,i 1,2 , for some “typical” values of ν
δ,b0,θ0 ∈{ }
Of course it would be interesting to study how fine and large the grid must be
in order that the CGEM-EV statistical performances be comparable to ML. One
may guess that the signal-to-noise ratio must also be “not too small”, even if its
value does not matter asymptotically. Since the main improvements in numerical
performance occur for multidimensional process (with randomized-traces used in-
stead of the exact traces in (1.3)), a first Monte Carlo study has been done for
two-dimensional random fields in Girard (2009).
This approch may be extended in a simple way to the case of unknown vari-
ance, says σ2, of the measurement error (or unknown nugget effect). Indeed it is
ε
well known (Stein (1999, Section 6.2)) that the simple average of local squared
differences like (y y )2 would yield here a consistent estimate σˆ2 , and this
i − i−1 ε
is easily generalized to multidimensional processes which are mean square contin-
uous. Notice that the Bayesian filter is still A except that b now denotes the
b,θ
signal-to-noise ratio s.t. var(Z) = bσ2. Then the whole CGEM-EV approach can
ε
be applied after having replaced y by its “standardized” version y/σˆ . It would
ε
be thus very interesting to study whether similar near-efficiency results still hold
or if a more elaborate estimate for σ is required.
ε
Concerning the case with known noise and signal variances, even if the as-
sumption that b were known is rather restrictive, it may be worth to point out
0
ˆ
the following very neat result: denoting by θ the ML estimate with b fixed at
ML0
b , its asymptotic behavior is classically obtained
0
1
n1/2 θˆ θ N 0,4π a2h2 − ,
ML0 − 0 →D 0 0 !
(cid:16) (cid:17) (cid:18)Z (cid:19)
ˆ
and denoting by θ a consistent solution of the estimating-equation (1.3) with
GE0
now b fixed at b , we have
0
Theorem 4.2. If a2h = 0 then
0 0 6
R 2
n1/2 θˆ θ N 0,4π a2 a2h −
GE0 − 0 →D 0 0 0 !
(cid:16) (cid:17) Z (cid:18)Z (cid:19)
and the ratio of this asymptotic variance over the one of θˆ , says I0 , satisfies:
ML0 δ,b0,θ0
I0 = 1+J (h ) 1 as δ 0.
δ,b0,θ0 0 0 → →
8
Asymptotic (i.e. large-n-small-δ) full-efficiency is thus now enjoyed by the
CGEM estimating function for any value of ν 1/2.
≥
ˆ
To finish, let us comment on the somewhat more natural b -based alterna-
EV
tive to ML which would be akin to the hybrid estimate proposed by Zhang and
ˆ ˆ
Zimmerman (2007) : choose θ so as to maximize the likelihood function with b
H EV
ˆ
plugged-in. As to the computational aspects, θ may be much less easily com-
H
puted than θˆ (for example when an iterative solver is used for A y) since
GEEV b,θ
it requires a log-determinant instead of a trace. So it is worth to point out that
again a neat result (which can be similarly obtained by the technics used to prove
ˆ
Theorem 4.1) holds true here: the large-n mean square error of θ over the one of
H
ˆ
θ tends to 1 as δ 0.
GEEV
→
5. Proofs
In the following, c denotes a constant that may change from line to line (that
is, c will only possibly depend on ν and on the positive lower and upper bounds
for the candidate b’s and those for θ (says b,¯b,θ,θ¯)). And O(δ2ν) denotes a term
whose absolute value is bounded by cδ2ν. Without loss of generality we assume
that δ 1.
≤
Firstly, it is easily checked from the definition of g (λ), that the un-aliased
ν∗,α
version of hδ has a very simple form:
ν,θ
λ 1
h (λ) = α 1 2ν (2ν +1)G , where G(ω) = . (5.1)
∗ν,α − − α 1+ω2
(cid:18) (cid:18) (cid:19)(cid:19)
We first collect intermediary results in a few lemmas which might be of interest
for other studies of this small-δ Mat´ern time-series model.
Lemma 5.1. We have
cδ2ν g (λ) 6 gδ (λ) 6 g (λ)+O(δ2ν),
≤ ν∗,δθ ν,θ ν∗,δθ
and
g (λ) hδ (λ) δh (λ) = O(δ2ν).
ν∗,δθ ν,θ − ∗ν,δθ
(cid:12) (cid:12)
(cid:12) (cid:12)
Lemma 5.2. We have
a (λ) 6 aδ (λ) 6 a (λ)+O(δ2ν),
∗b,δθ b,θ ∗b,δθ
9
and
aδ (λ) 6 cg (λ).
b,θ ν∗,bθ
Proof of these Lemmas. From their definition, we have g (λ) g (λ) =
ν,α − ν∗,α
g (λ+2πk). For any λ [ π,π], for any k 1, the monotonicity of g
k=0 ν∗,α ∈ − ≥ ν∗,α
imp6lies g (λ + 2πk) g (2π(k 1/2)) cα2ν/(k 1/2)2ν+1; and summing
P ν∗,α ≤ ν∗,α − ≤ −
these terms over k = 1,2, , thus gives a term O(α2ν). This is shown similarly
···
for the sum over k = 1, 2, . We now prove the second part of Lemma 5.1.
− − ···
Omitting the index ν and denoting g˙ := ∂g /∂α, g˙ := ∂g /∂α, one can decom-
α α α∗ α∗
pose αg δ 1hδ h = αg (g˙ /g g˙ /g ) = (g g )(αg˙ /g )+α(g˙ g˙ ).
α∗ − θ − ∗α α∗ α α − α∗ α∗ α∗ − α α α α − α∗
Observing that αg˙ = 2νg + cg yields α(g˙ g˙ ) = O(α2ν). Lastly,
(cid:0) α∗ (cid:1) ν∗,α ν∗+1,α α − α∗
we can bound α g˙ /g α g˙ /g α g˙ /g + α g˙ g˙ /g O(1) +
| α α| ≤ | α α∗| ≤ | α∗ α∗| | α − α∗| α∗ ≤
α g˙ g˙ /cα2ν from equation (5.1) and the first inequality of Lemma 5.1.
| α − α∗|
Lemma 5.3. If ν > 1/2, then for any continuous function F : R R such that
+
→
0 < ∞ F(x)dx < + , we have, as δ 0
∞ →
−∞
R
λ λ
aδ (λ)F dλ δθ ∞ F(x)dx, aδ (λ) 2F dλ δθ ∞ F(x)dx.
b,θ δθ ∼ b,θ δθ ∼
Z (cid:18) (cid:19) Z Z (cid:18) (cid:19) Z
−∞ −∞
(cid:2) (cid:3)
Proof. The first part of Lemma 5.2 and the boundedness of F permits us to
replace aδ by themuch moremanageablea (where α = δθ)with anerrorO(δ2ν)
b,θ ∗b,α
whichisenoughwhenν > 1/2. Lettingλ¯ = αǫ+(2ν/(2ν+1)) whereǫisanyarbitrarily
small constant > 0, we see from the monotonicity of g , that inf a (λ)
1 ν∗,α λ∈[−λ¯,λ¯] ∗b,α ≥
1+(2π/b) gν∗,α(λ¯) −1 − which tends to 1 since gν∗,α(λ¯) ≥ α2ν/ 2λ¯2 ν+1/2 (from
α(cid:16) small eno(cid:2)ugh) wh(cid:3)ich(cid:17)tends to +∞. And the same is true fo(cid:0)r a(cid:1)∗b,α(λ) 2. It
remains to observe that both π F λ dλ and λ¯ F λ dλ are equivalent to the
π δθ λ¯ δθ (cid:2) (cid:3)
claimed term. − −
R (cid:0) (cid:1) R (cid:0) (cid:1)
Lemma 5.4. We have
aδb,θ(λ)dλ ∼ (δθ)2ν2+ν1(2πcνb)2ν1+1 ∞ (1+x2ν+1)−1dx,
Z Z
−∞
aδb,θ(λ) 2dλ ∼ (δθ)2ν2+ν1(2πcνb)2ν1+1 ∞ (1+x2ν+1)−2dx.
Z Z
−∞
(cid:2) (cid:3)
Proof. By Lemma 5.2 we can replace, with enough accuracy, the filter by its un-
aliased version in these integrals. Next the claimed equivalents can be obtained by
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