variance estimation for fractional brownian … · for peer review only variance estimation for...

For Peer Review O

nly

VARIANCE ESTIMATION FOR FRACTIONAL BROWNIAN

MOTIONS WITH FIXED HURST PARAMETERS

Journal: Communications in Statistics – Theory and Methods

Manuscript ID: LSTA-2011-0573.R1

Manuscript Type: Original Paper

Date Submitted by the

Author: n/a

Complete List of Authors: Coeurjolly, Jean-Francois; Laboratory Jean Kuntzmann, Department of Statistics Lee, Kichun; Hanyang University, Industrial Engineering Vidakovic, Brani; Georgia Institute of Technology,

Keywords: fractional Brownian motion, variance estimation, Hurst exponent, turbulence signals

Abstract:

Some real-world phenomena can be effectively modeled by a fractional Brownian motion indexed by a Hurst parameter, a regularity level, and a scaling parameter $\sigma^2$, an energy level. This paper discusses estimation of $\sigma^2$ when a Hurst parameter is known. To estimate $\sigma^2$, we propose three

approaches based on maximum likelihood estimation, moment matching, and concentration inequalities, respectively, and discuss the theoretical characteristics of the estimators and optimal-filtering guidelines. In a simulation study, we compare the confidence intervals of $\sigma^2$ in terms of their lengths, coverage rates, and computational complexity and discuss empirical attributes of the three approaches.

URL: http://mc.manuscriptcentral.com/lsta E-mail: [email protected]

Communications in Statistics ? Theory and Methods

For Peer Review O

nly

of 26



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

VARIANCE ESTIMATION FOR FRACTIONAL BROWNIAN MOTIONS WITH FIXED

HURST PARAMETERS

Jean-Francois Coeurjolly1, Kichun Lee2, Brani Vidakovic3

1 Laboratory Jean Kuntzmann, Department of Statistics

Grenoble University, France

[email protected]

2 Biomarkers Laboratory, Department of Medicine

Emory University, Atlanta GA

[email protected]

3 Department of Biomedical Engineering

Georgia Institute of Technology, Atlanta GA

[email protected]

Key Words: fractional Brownian motion; variance estimation; Hurst exponent; turbulence

signals.

ABSTRACT

Some real-world phenomena in geo-science, micro-economy, and turbulence, to name a

few, can be effectively modeled by a fractional Brownian motion indexed by a Hurst param-

eter, a regularity level, and a scaling parameter σ2, an energy level. This paper discusses

estimation of a scaling parameter σ2 when a Hurst parameter is known. To estimate σ2,

we propose three approaches based on maximum likelihood estimation, moment-matching,

and concentration inequalities, respectively, and discuss the theoretical characteristics of the

estimators and optimal-filtering guidelines. We also justify the improvement of the estima-

tion of σ2 when a Hurst parameter is known. Using the three approaches and a parametric

bootstrap methodology in a simulation study, we compare the confidence intervals of σ2 in

terms of their lengths, coverage rates, and computational complexity and discuss empiri-

1

of 26



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

cal attributes of the tested approaches. We found that the approach based on maximum

likelihood estimation was optimal in terms of efficiency and accuracy, but computationally

expensive. The moment-matching approach was found to be not only comparably efficient

and accurate but also computationally fast and robust to deviations from the fractional

Brownian motion model.

1. INTRODUCTION AND MOTIVATION

A wide range of complex structures in natural phenomena exhibit fractal properties such

as self-similarity and long-range dependence. To model such properties, a fractional Brow-

nian motion (fBm), as a universal scaling model, has been widely used since the pioneering

work of Mandelbrot and Van Ness (1968). The fractional Brownian motion can be defined

as the unique centered Gaussian process, denoted by (X(t))t≥0, with stationary increments

and self-similarity property: for all c > 0, X(ct)d= cHX(t), where

d= is the equality in all

finite-dimensional distributions. Its variance function given by V arX(t) = σ2|t|2H is param-

eterized by a scaling constant σ > 0 and a parameter H ∈ (0, 1) often referred to as the

Hurst exponent. The Hurst parameter H characterizes the path regularity of a fBm process

since the fractal dimension of paths is equal to D = 2 −H . Thus, sample paths with small

H are “space filling”, while paths with H close to 1 are almost smooth. The scaling constant

σ determines the energy level of the process.

The discretized increments of a fBm (called fractional Gaussian noise) constitute a short-

range dependent process, when H < 1/2, and a long-range dependent process, when H >

1/2. General references on self-similar and long-memory processes are given in Beran (1994)

or Doukhan et al. (2003). The power spectrum P for a fBm is given by the power law

P ∝ |f |−β over some range of frequencies f , where the spectral exponent β relates to the

Hurst parameter H as H = β−12. Thus, the empirical spectral exponent can be used to

estimate H . Besides the power spectrum based approach, a variety of estimators of the

Hurst parameterH exists in the literature. The reader is referred to Beran (1994), Coeurjolly

(2000), or Bardet et al. (2003) for an overview of this problem. Nevertheless, estimation of

the scaling constant σ, mostly treated as a nuisance parameter, has received little attention

2

of 26



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

from the statistics community. Veitch and Abry (1999) provided the identification of both the

Hurst parameter H and the scaling constant σ by regressing the logarithm of the scalograms

(i.e., empirical variance of the wavelet transform of the data at a given scale) against the

range of scales. Coeurjolly (2001) also considered the joint estimation of H and σ2 using

discrete filtering techniques instead of wavelet techniques. If σ2 is of interest and H is known,

the estimation of σ2 can be improved.

Indeed, fBm models with known Hurst parameters can effectively describe some real prob-

lems in modern engineering disciplines. Among those problems, best known are turbulence

processes formulated by Kolmogorov’s K41 theory for isotropic turbulence (Kolmogorov,

1941). The power spectrum P for locally isotropic turbulence is expressed by P ∝ |f |−5/3,

which is known as Kolmogorov’s “5/3” law, leading to the Hurst parameter H = 1/3. In

particle systems, it has been known that the limiting behavior of some colliding Brownian

motions on the real line converges to a fBm with a fixed Hurst parameter H = 1/4 (Nourdin

and Reveillac, 2009; Peligrad and Sethuraman, 2008; Swanson, 2010). In DNA sequences,

Arneodo (1996) showed that non-coding regions of human genes expressed as DNA “ran-

dom walk” have Hurst parameters close to 0.6. In atmospheric turbulence, wave fronts that

have passed through a turbulent atmosphere become a homogeneous and isotropic stochastic

Gaussian process and are identified as a fBm with the Hurst parameter H = 5/6 (Schwartz

et al., 1994; Ribak, 1997; Perez et al., 2004). Moreover, several sophisticated turbulence

models (Nelkin, 1975; Biskamp, 1994; Horbury et al., 2008) have been presented that yield a

power law |f |−β with β different from 5/3. Under the fBm model, this power law can result

in several fixed Hurst parameters. Consequently, the ample evidence of such real problems

makes it relevant to correctly identify a fBm with a fixed Hurst parameter.

Thus, the objective of this paper is to study estimation of σ2 when the Hurst parameter is

known. To derive interval estimators of σ2, we discuss several approaches that are based on

maximum likelihood estimation, moment-matching accompanied by the central limit theorem

(CLT), and concentration inequalities, which is provided in Section 3. Besides, we provide

an efficiency comparison and optimal-filtering guidelines for the methods. Using asymptotic

3

of 26



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

properties of the estimators, we also justify the improvement of the σ2 estimation with the

knowledge of H . By running a simulation study, we construct confidence intervals of σ2 by

the three approaches as well as a parametric bootstrap approach as a baseline, which are

provided in Section 4. We also compare the performance of those approaches in terms of

covering rates, the length of confidence intervals, and computational time and discuss the

observed characteristics of the methods.

2. BACKGROUND AND NOTATION

We recall that a fractional Brownian motion (fBm) can be defined as the unique centered

Gaussian process, denoted by (X(t))t≥0, with stationary increments. Its variance function

given by V arX(t) = σ2|t|2H is parameterized by a scaling constant σ > 0 and a Hurst pa-

rameter H ∈ (0, 1). As a direct application of the stationarity and self-similarity properties,

we obtain the covariance function of the fBm as

EX(s)X(t) =σ2

2

(|s|2H + |t|2H − |t− s|2H

), s, t ≥ 0.

From now on, X will denote the vector of observations at times i/n for i = 0, . . . , n − 1 of

the fBm. We consider a filter a of length ℓ + 1 and order p. That means that ℓ + 1 filter

taps ai, 0 ≤ i ≤ ℓ, satisfy

ℓ∑

q=0

qjaq = 0 for j = 0, . . . , p− 1 and

ℓ∑

q=0

qjaq 6= 0 for j ≥ p. (1)

For instance, we shall consider the following filters: increments i1 = (−1, 1) with ℓ = 1,

p = 1, increments 2 i2 = (1,−2, 1) with ℓ = 2, p = 2, Daublet 4 d4 = (−0.09150635,

−0.15849365, 0.59150635, −0.34150635) with ℓ = 3, p = 2, Coiflet 6 c6 = (−0.05142973,

−0.23892973, 0.60285946, −0.27214054, −0.05142973, 0.01107027) with ℓ = 5, p = 2 (see

e.g. Daubechies (2006) and Percival and Walden (2000) for more details). We will also

consider i3 = (1,−3, 3,−1) and c12 which is Coiflet 12 with ℓ = 12, p = 4. Let Xa denote

the vector X filtered by a with components

Xa

(i

n

):=

ℓ∑

q=0

aqX

(i− q

n

), for i = ℓ, . . . , n− 1.

4

of 26



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

Let πa

H(·) and ρaH(·) be, respectively, the covariance and the correlation functions of the

filtered series. Using basic properties of the fBm, this covariance function can be represented

as (see e.g. Coeurjolly, 2001)

E[Xa(k/n)Xa((k + j)/n)] =σ2

n2H× πa

H(j) with πa

H(j) = −1

2

ℓ∑

q,r=0

aqar|q − r + j|2H (2)

and ρaH(·) := πa

H(·)/πa

H(0). As an example, when a = i1, Xa corresponds to the increments

of the fBm, which is fractional Gaussian noise (fGn), equation (2) recovers the covariance of

the fGn as

E[X i1(k/n)X i1((k + j)/n)] =σ2

2n2H

(|j − 1|2H − 2|j|2H + |j + 1|2H

). (3)

The interest in filtering a fractional Brownian motion is motivated by the following result:

for any filter of order p ≥ 1, Xa is a stationary sequence and as |i| → +∞,

ρaH(i) ∼ c|i|2H−2p, (4)

where c is a constant depending only on H and a (see e.g. Lemma 1 of Coeurjolly, 2001), but

not on i. Therefore, the higher p is, the more decorrelated the resulting filtered observations

are. This is an important remark since the confidence intervals of σ2 we propose will require

the square or absolute summability of ρaH(·).

3. CONFIDENCE INTERVALS OF σ2 WHEN HURST PARAMETERS ARE FIXED

Next we discuss the inference on σ2 in the case of known H . At first glance the problem

may look trivial since the process is Gaussian and with known H the dependence structure in

the sample paths is fully known. Indeed, the maximum likelihood estimator of σ2 is straight-

forward. However, it is well known that unlike location parameter cases, the estimators of

scale parameters are quite sensitive to deviations from the model. This motivates introduc-

tion of two additional robust counterparts, moment-matching and concentration inequality

based estimators. As we will see later, the efficiency of the moment-matching estimator is

comparable to that of the maximum likelihood estimator and is more robust to deviations

from the fBm model. In addition to efficiency and robustness, the benefit of introducing

5

of 26



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

those estimators is in computational complexity. The maximum likelihood estimator re-

quires inversion of a Toeplitz matrix, typically of a large size, while the newly introduced

estimators are obtained by fast filtering. In all cases interval estimators of σ2 are derived

and assessed.

3.1. Maximum likelihood estimation (MLE)

In this section, we consider the maximum likelihood estimator of σ2 based on the observa-

tions of fGn, that is on Xi1 = (X i1(i/n))i=1,...,n−1. The following result states the maximum

likelihood estimate of σ2 and its confidence interval.

Proposition 1 The maximum likelihood estimate of σ2 is given by σ2mle :=

1n−1

Xi1TR−1Xi1,

where R is the Toeplitz covariance matrix of standard fGn (σ2 = 1) with elements given

by (3). The σ2mle satisfies (n− 1)

σ2

mle

σ2 ∼ χ2(n− 1). Hence, given α ∈ (0, 1/2)

P

((n− 1)σ2

mle

q1−α/2≤ σ2 ≤ (n− 1)σ2

mle

qα/2

)= 1− α, (5)

where qα is the α-quantile of a χ2(n− 1) distribution.

The proof is straightforward due to the fact that H is known and the process is Gaussian.

We will discuss the variational property of the MLE-based estimator in comparison with the

following moment-matching estimator.

3.2. Moment-matching approach (CLT)

In this section we propose a moment-matching estimator of σ2 and explore its asymptotic

properties. Let Sa

n and V a

n be defined as

Sa

n :=1

n− ℓ

n−1∑

i=ℓ

Xa

(i

n

)2

and

V a

n :=1

n− ℓ

n−1∑

i=ℓ

((Xa(i/n)

E(Xa(i/n)2)1/2

)2

− 1

)=

n2HSa

n

σ2πa

H(0)− 1.

The statistics V a

n can thus be viewed as the empirical mean of the H2−variations (where

H2(t) = t2 − 1 is the second Hermite polynomial) of a stationary Gaussian sequence with

6

of 26



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

covariance function ρaH . Since, from (2), E[Sa

n] = σ2πa

H(0)/n2H and H is assumed known,

the following estimator of the variance is derived

σ2a:=

n2HSa

n

πa

H(0). (6)

For example, for a = i1 or i2, we have simple expressions πi1

H (0) = 1 and πi2

H (0) = 4 − 22H .

Note the following important relation

σ2a

σ2=

n2HSa

n

σ2πa

H(0)= 1 + V a

n . (7)

Therefore, asymptotic properties of σ2aare readily deduced from properties of V a

n established

in Coeurjolly (2001).

Proposition 2

(i) As n → +∞, σ2aconverges towards σ2 almost surely.

(ii) If the order of filter a satisfies p > H+1/4, then the following convergence in distribution

holds as n → +∞√n− ℓ

(σ2a

σ2− 1

)d−→ N

(0, τ 2

), (8)

where τ 2 := τ 2(H, a) = 2‖ρaH‖2ℓ2(Z) with ‖ρaH‖2ℓ2(Z) =∑

i∈Z ρa

H(i)2.

(iii) For α ∈ (0, 1/2) and n such that√n− ℓ > τqα/2,

P

(σ2a

1 + τqα/2/√n− ℓ

≤ σ2 ≤ σ2a

1− τqα/2/√n− ℓ

)−→ 1− α, n → +∞, (9)

where qα = Φ−1(1− α) with Φ the cumulative distribution function of N (0, 1).

Proposition 1 of Coeurjolly (2001) asserts the almost sure convergence of V a

n to 0 and the

convergence in distribution of√n− ℓ V a

n towards a Gaussian distribution with zero mean

and variance τ 2. Therefore Proposition 2 (i) and (ii) are easily deduced. The result (iii)

is a consequence of the convergence in distribution in (8) which allows us to construct an

asymptotic confidence interval for σ2.

The condition p > H+1/4 is related to the square summability of the correlation function

ρaH . According to (4), a filter of order p ≥ 2 induces no requirement on the Hurst parameter

H while choosing a filter with order p = 1 implies that (8) holds only for H ∈ (0, 3/4).

7

of 26



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

The asymptotic constant τ 2(H, a) cannot be analytically computed. However, numerical

approximations can easily be undertaken. We have computed these approximations for a

large number of different filters. A part of the results is shown in Figure 1. For filters with

order p ≥ 2, we observe that the constant ‖ρaH‖ℓ2(Z) grows with p, which means that it is

not so beneficial to strongly decorrelate the observations since small variances in (8) are

desirable. This constant also becomes quite large when H is getting close to 3/4 for the

differencing filter i1. From this study, the optimal filter is the filter i1 when H ∈ (0, Hc] and

the filter d4 for H ∈ [Hc, 1), where the critical value Hc, numerically computed, is evaluated

to .641.

The moment-matching approach (also referred to as the central limit theorem (CLT)

approach in the sequel) and the MLE approach can be compared using asymptotic efficiency

as in Figure 1. Indeed, from Proposition 1, V ar(√

n− 1σ2

mle

σ2

)∼ 2, and therefore the efficiency

of the MLE estimate compared to the moment-matching one is given by

eff(σ2mle, σ

2a) =

V ar(σ2a)

V ar(σ2mle)

∼ ‖ρaH‖2ℓ2(Z).

Figure 1 shows that the MLE based estimate is more efficient than the moment-matching

estimate since the lower bound of ‖ρaH‖ℓ2(Z) is approximately 1 for all the tested filters. It also

represents that, when the optimal filter guideline is applied, the MLE based estimate becomes

quite comparable with the moment-matching estimate since its efficiency is no greater than

1.222 ≃ 1.488 for the filter i1 and H close to 0.

8

of 26



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

Figure 1: Evolution of ‖ρaH‖ℓ2(Z) in terms of H for different filters a. The infinite series are

truncated to the set of indices −300, . . . , 300.

3.3. Approach based on concentration inequalities (CI)

What could appear to be a drawback in the result (9) is that the confidence interval

of the variance σ2 is asymptotic. In this section, we consider a related approach based on

concentration inequalities. With the notation of the present paper, Breton and Coeurjolly

(2010) obtained the following inequalities for the tails of the random variable V a

n . These

inequalities are improvements of concentration inequalities obtained by Breton et al. (2009).

Lemma 3 Let a be a filter of order p > H + 1/2 and let κ := κ(H, a) = 2‖ρaH‖ℓ1(Z), then

9

of 26



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

for all t > 0

P(√

n− ℓ V a

n ≥ t)

≤ ϕr,n(t) := e− tn

κ√

n−ℓ

(1 +

t√n− ℓ

)n

κ

P(√

n− ℓ V a

n ≤ −t)

≤ ϕl,n(t) := etn

κ√

n−ℓ

(1− t√

n− ℓ

)n

κ

1[0,√n−ℓ](t).

The condition p > H + 1/2 is related to the absolute summability of ρaH . This condition

is always fulfilled for any filter of order p ≥ 2, whereas it is valid for H ∈ (0, 1/2] only

when p = 1. Breton and Coeurjolly (2010) obtained explicit values of ‖ρaH‖ℓ1(Z) for filters acorresponding to the increments:

‖ρi1H‖ℓ1(Z) = 2 for all H ∈ (0, 1/2)

‖ρi11/2‖ℓ1(Z) = 1

‖ρi2H‖ℓ1(Z) =

1 + 10−7×4H+2×9H

4−4Hif H < 1/2

2 if H ≥ 1/2.

For other filters like Daubechies wavelet filters or Coiflet filters, a numerical approximation

has to be made. Functions ϕr,n and ϕl,n depend on filter a through constant κ. To optimize

the inequalities in Lemma 3, the smallest κ, i.e. the smallest ‖ρaH‖ℓ1(Z), are of particular

interest. From the evolution of ‖ρaH‖ℓ1(Z) in terms of H for different filters a as shown in

Figure 2, the optimal filter is i1 for H ∈ (0, 1/2] and i2 or d4 for H ∈ (1/2, 1).

10

of 26



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

Figure 2: Evolution of ‖ρaH‖ℓ1(Z) in terms of H for different filters a. The infinite series are

truncated to the set of indices −300, . . . , 300. For convenience, ‖ρi1H‖ℓ1(Z) is not shown for

H = 1/2 (recall that it equals 1 for this value).

Lemma 3 was used in Breton and Coeurjolly (2010) to derive confidence intervals of the

Hurst parameter H independently of σ2. Since we are interested in the parameter σ2 with

H being fixed, we obtain the following confidence interval.

Proposition 4 Let us define q•,α = ϕ−1•,n(α) for α ∈ (0, 1/2) and • = l, r, then for all n > ℓ,

we have

P

(σ2a

1 + qr,α/2/√n− ℓ

≤ σ2 ≤ σ2a

1− ql,α/2/√n− ℓ

)≥ 1− α. (10)

11

of 26



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

Proof. Lemma 3 entails that

P(−ql,α/2 ≤

√n− ℓ V a

n ≤ qr,α/2

)= P

(√n− ℓ V a

n ≤ qr,α/2

)− P

(√n− ℓ V a

n ≤ −ql,α/2

)

≥ 1− α

2− α

2= 1− α,

which is, by (7), equivalent to

P

(−ql,α/2 ≤

√n− ℓ

(σ2a

σ2− 1

)≤ qr,α/2

)≥ 1− α.

Equation (10) is then easily deduced.

The confidence interval (9) is valid for n → +∞, which means that the 1− α confidence

level is asymptotically achieved. Unlike the MLE interval, equation (10) implies that, for

n > ℓ, the confidence level is at least 1− α.

3.4. Improvement of the estimation of σ2 when Hurst parameters are fixed

In this section, we develop a remark in the introduction: the estimation of σ2 is improved

when H is known. To justify it, we describe asymptotic properties of σ2 estimators both

when H is known and when H is unknown. Then we examine accuracy of both confidence

intervals of σ2 in terms of the mean squared error (mse) by a simulation study. For this

purpose, we focus on the moment-matching estimator in Section 3.2 for known H and the

joint estimator of (H, σ2) in Coeurjolly (2001) for unknown H . Similar comments, however,

can be stated for the the MLE-based estimator since the asymptotic efficiency of the moment-

matching estimator is quite comparable with that of the MLE-based estimator as described

above.

For unknown H , the joint estimation of (H, σ2) in Coeurjolly (2001), using discrete

variations, simply consists in plugging an estimate of H in (6). Let H be an estimate of H

converging almost surely towards H (as n → +∞) and satisfying a central limit theorem

with rate√n: i.e., there exists ωH > 0 such that

√n(H − H) converges in distribution

towards a centered Gaussian variable with variance ω2H. Let σ2

a,Hstand for the estimate

defined by (6) where H is replaced by H . Then, similarly as Proposition 2 of Coeurjolly

12

of 26



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

(2001) shows, Proposition 2 and Slutsky’s theorem may be combined to prove that

√n− ℓ

2 log(n)

(σ2a,H

σ2− 1

)d−→ N

(0, ωH

). (11)

Compared to (8), the absence of known H in the σ2 estimation incurs an additional fac-

tor of order log(n) that slows the convergence in distribution. We confirm this inefficiency

by empirical findings, as shown in Figure 3. The simulation was based upon 500 replica-

tions of fBms with Hurst parameter H = 1/3 and H = 5/6 with variance σ2 = 1. For

known H , we used the R package dvfBm, which implements the joint estimator in Coeurjolly

(2001), applying the optimal filter guideline and dilating the optimal filter up to 5 times.

For instance, when H = 1/3 in Figure 3(a) (and H = 5/6 in Figure 3(b)), the logarithmic

difference of the mse between the two estimates is at least greater than 4 (respectively, 5.7)

for log(n) = 5, . . . , 10. It demonstrates that the knowledge of H improves considerably the

performance of the σ2 estimation.

13

of 26



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

(a) H = 1/3, σ2 = 1 (b) H = 5/6, σ2 = 1

Figure 3: Mean squared errors of variance estimates defined by σ2ain (6) for known H (in

circles) and by σ2a,H

in (11) for unknown H (in triangles) in terms of the logarithmic scales of

n. For the latter, the plugged estimate of H (H) is based on discrete variations (Coeurjolly,

2001). The simulations are based on 500 replications of fBms with Hurst parameter (a)

H = 1/3 and (b) H = 5/6 with variance σ2 = 1.

4. SIMULATION STUDY AND DISCUSSION

In this section, we compare the confidence intervals of σ2 based on maximum likelihood

estimation, moment-matching, and the concentration inequalities as in (5), (9), and (10),

respectively, through a simulation study. These methods are referred to as mle, clt, and ci,

respectively. We also consider a parametric bootstrap approach consisting of the following

steps:

1. Estimate the variance of a given fBm by (6) and denote it by σ2.

2. Generate B (e.g. B = 500) replications of a fBm with Hurst parameter H and with

variance σ2.

14

of 26



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

3. Estimate the variances on each replicated fBm and denote them by σ21, . . . , σ

2B.

4. Define the bootstrap (1− α) confidence interval for σ2 by[σ2([Bα/2]), σ

2([B(1−α/2)])

].

The motivation for including a parametric bootstrap approach is that, based on parametric

assumptions, it is straightforward to simulate empirical null-distributions of σ2 as an alter-

native to the construction of confidence intervals. This last method is referred to as boot in

the results and serves as a baseline in comparison with the three analytical methods above.

This simulation study has been done using the R software, and codes are available from the

authors on demand.

Tables 1 and 2 for n = 50, 100 and Tables 3 and 4 for n = 500, 5000 present, respectively,

the empirical coverage rates and the mean lengths of the confidence intervals, based on

m = 500 replications of sample paths of a fBm for selected pairs of parameters. The coverage

rate of the confidence intervals for one method was defined to a ratio of the number of

confidence intervals that included the known variance parameter to the total number of

replications (m = 500). We remark that the mle, clt, and boot perform well and in a

quite similar way. These three methods mostly respect the expected confidence level of

1 − α = 95%. In particular for small n, mle and boot performed well. In terms of the

mean length, the best method is without any surprise the mle, but both clt and boot

come close. Although being characterized by its non-asymptoticity, the ci method stands

out by its noticeable poor efficiency, whether n is small or big. This method, being clearly

too conservative, makes the resulting intervals quite large: the mean length of confidence

intervals obtained by the ci method is more than twice the mean lengths of confidence

intervals obtained by the other three methods.

From the computational point of view, the best method is, without any doubt, the clt

method which is extremely fast (even for sample sizes of order n = 105). The mle method

requires the computation of the inverse of an n × n Toeplitz matrix. For the Monte-Carlo

simulations, this has to be computed only one time since this matrix R depends only on the

known Hurst parameter H . Then the evaluation of the m = 500 mle estimates is reasonably

fast. Nevertheless, we note that n = 5000 is a limitation for the use of the mle method

15

of 26



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

on a basic R installation. The boot method is also very slow: each replication requires

B = 500 simulations of a sample path of length n and re-estimation of the scale parameter.

This was quite expensive for n ≥ 5000. Table 5, presenting the time in seconds required

to compute the confidence interval for one replication, illustrates the above comments: the

clt method was the fastest and the mle method, becoming exponentially slow with n, was

computationally unfeasible for n = 50000 on a basic R installation.

Coverage rate of the confidence intervals (%)

n = 50 n = 100

Parameters mle clt ci boot mle clt ci boot

σ = .1, H = 1/3 94.8 95.2 99.8 94.6 94.6 93.6 100 93.8

H = 3/5 95.8 93.4 100 90.2 95.6 94.8 100 94.4

H = 5/6 94.2 94.4 99.8 95.0 95.80 94.6 100 94.6

σ = 1, H = 1/3 94.8 95.6 100 96 95.6 94.2 100 95.6

H = 3/5 95.4 93.2 100 94 95.4 96 100 93.8

H = 5/6 93 95 99.8 94.2 93.8 95.6 100 96

σ = 10, H = 1/3 94.2 95.6 99.8 96.2 94 95 100 94.6

H = 3/5 94.4 94.2 100 94 95.6 93.2 100 93.4

H = 5/6 94.2 93.8 99.6 94 95.8 94.6 100 94.6

Table 1: Empirical coverage rate of the confidence intervals for the mle, clt, ci, boot

methods based on m = 500 replications of a fBm for selected pairs of parameters and for

n = 50 and 100. The coverage rate, being a ratio of the number of confidence intervals

that included σ2 to m, represents the accuracy of one method. The filter a used for the

methods clt, ci, and boot depends on the value of H and corresponds to the optimal filter

combination discussed in Sections 3.2 and 3.3.

16

of 26



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

Normalized mean length of the confidence intervals

n = 50 n = 100


σ = .1, H = 1/3 86.8 93.1 195.6 83.6 57.9 64.3 122.9 58.7

(×104) H = 3/5 88.0 93.3 201.5 90.7 58.5 63.1 124.9 63.1

H = 5/6 88.9 93.5 205.1 93.0 59.1 66.9 124.5 61.7

σ = 1, H = 1/3 85.3 93.5 196.6 84.1 58.2 64.2 122.7 58.5

(×102) H = 3/5 87.8 95.1 202.3 91.1 58.8 63.4 123.6 62.6

H = 5/6 88.6 94.6 207.2 93.9 59.0 67.6 125.8 62.4

σ = 10, H = 1/3 86.7 93.2 195.9 83.8 57.8 63.6 121.6 57.9

H = 3/5 87.8 93.8 202.5 91.2 59.2 63.7 123.4 62.1

H = 5/6 89.3 95.4 208.6 94.4 58.9 67.4 125.3 62.1

Table 2: Mean length of the confidence intervals for the setup as in Table 1.

17

of 26



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly


n = 500 n = 5000


σ = .1, H = 1/3 93.0 94.4 100 93.6 96.6 94.8 100 93.8

H = 3/5 94.4 95.4 100 92 96.4 94.2 99.8 94.4

H = 5/6 93.4 94.4 100 94 93.6 97.2 100 96.6

σ = 1, H = 1/3 95.8 94 100 93.6 95 94.8 100 94.4

H = 3/5 92.8 95.8 100 94.6 94.4 93.8 100 95.6

H = 5/6 93.4 95 100 94.6 95.8 94.6 100 94.4

σ = 10, H = 1/3 95.4 94.4 100 93.6 95.8 95.2 100 94.8

H = 3/5 96.2 96.2 99.8 93.4 94.6 94.6 100 93.8

H = 5/6 94.6 94.2 100 93.8 93.8 95.6 100 94.8

Table 3: Empirical coverage rate of the confidence intervals for the mle, clt, ci, boot

methods based on m = 500 replications of a fBm for selected pairs of parameters and for

n = 500 and 5000. The coverage rate, being a ratio of the number of confidence intervals

that included σ2 to m, represents the accuracy of one method. The filter a used for the

methods clt, ci, and boot depends on the value of H and corresponds to the optimal filter

combination discussed in Sections 3.2 and 3.3.

18

of 26



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

Normalized mean length of the confidence intervals

n = 500 n = 5000


σ = .1, H = 1/3 25.0 26.4 49.6 25.7 7.9 8.2 15.4 8.1

(×104) H = 3/5 25.0 26.4 49.9 27.3 7.8 8.2 15.4 8.6

H = 5/6 25.1 27.2 50.1 26.6 7.9 8.4 15.4 8.3

σ = 1, H = 1/3 25.0 26.4 49.7 25.8 7.9 8.2 15.4 8.1

(×102) H = 3/5 25.0 26.4 49.8 27.3 7.9 8.2 15.4 8.6

H = 5/6 25.2 27.1 49.8 26.5 7.8 8.4 15.4 8.3

σ = 10, H = 1/3 25.0 26.4 49.8 25.8 7.8 8.2 15.4 8.1

H = 3/5 25.1 26.3 49.7 27.3 7.8 8.2 15.4 8.6

H = 5/6 25.1 27.2 50.0 26.6 7.8 8.4 15.4 8.3

Table 4: Mean length of the confidence intervals for the setup as in Table 3.

n = 50 n = 500 n = 5000 n = 50000

mle 0.004 0.14 94.5 ×clt 0.19 0.18 0.18 0.19

ci 0.02 0.18 1.83 18.1

boot 0.25 0.59 6.02 60.33

Table 5: Time in seconds required to evaluate one confidence interval for the methods in

terms of n. The cross (×) means that the mle method was computationally unfeasible on a

basic R installation for such a sample size.

Furthermore, we compare the robustness of mle and clt methods since it is desirable

that the estimators are resilient to incidental violation of the assumed model. As a method

based on the likelihood function, Proposition 1 is specific to the fBm model. Unlike this,

19

of 26



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

Coeurjolly (2008) proved that the same convergence of the statistics V a

n can be obtained

for stationary Gaussian processes which exhibit a local self-similarity in 0. Such a Gaussian

process is defined by variance function of the form

v(t) = EX(t)2 = σ2|t|2H(1 + r(t)), with r(t) = o(1) as |t| → 0,

for some 0 < H < 1. In particular (see Coeurjolly, 2008 Theorem 4), if the remainder term

r(t) satisfies r(t) = O(|t|η) with η > 1/2 , then the convergence in distribution still holds

and therefore the conclusion of Proposition 2 still applies, which means that the confidence

interval derived by the clt method is still theoretically justified. We selected three examples

of functions v(t), 1− exp(−|t|2H), log(1 + |t|2H), and 2− 2 cos(|t|H) with r(t) = O(|t|η) andη = 2H > 1/2 that satisfy the above criterion. The coverage rates for the two methods

for different pairs of parameters are shown in Table 6, where the clt method consistently

surpassed the mle method. This result demonstrates high sensitivity to small deviations

from the fBm model of the mle method with respect to the clt method.

20

of 26



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly


1− exp(−|t|2H) log(1 + |t|2H) 2− 2 cos(|t|H)Parameters mle clt mle clt mle clt

σ = .1, H = 1/3 88.5 95.2 89.3 95.5 88.5 94.9

H = 3/5 90 95.1 88.9 95 88.8 95

H = 5/6 88.7 94.7 89.5 94.6 88.3 94.9

σ = 1, H = 1/3 88.4 94.4 88.5 94.8 89.6 95.5

H = 3/5 89.3 94.9 89.4 95.1 88.6 95

H = 5/6 88.5 94.9 88.8 94.7 89.4 95.2

σ = 10, H = 1/3 88.6 94.5 89.3 95.5 89.4 95

H = 3/5 88.8 94.7 88.7 94.9 89.3 94.6

H = 5/6 89 94.9 88.8 94.5 89.4 94.9

Table 6: Empirical coverage rate of the confidence intervals for the mle and clt methods

based on m = 500 replications of sample paths of length n = 500 of locally self-similar

Gaussian processes for the selected sets of parameters. Filter a used for the method clt

depends on the value of H and corresponds to the optimal-filtering guideline discussed in

Section 3.2. In our simulations, we used examples v(t) = |t|2H(1 + r(t)) with r(t) = O(|t|η)with η = 2H > 1/2 for our selection of H .

5. CONCLUSION

In this paper, we proposed three interval estimators of σ2 in the model of a fBm when the

parameterH is known and provided an efficiency comparison and optimal-filtering guidelines.

We also justified that the knowledge ofH improves the σ2 estimation, using asymptotic prop-

erties of the estimators. Then we compared the performance of the three estimators including

a parametric bootstrapping approach via a simulation study. Although the MLE-based esti-

mator among them is the optimal in the sense of efficiency, coverage rates, and mean lengths,

it is specific to fBm models and computationally expensive. We found that the moment-

21

of 26



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

matching estimator achieves comparable efficiency and is computationally inexpensive and

robust to deviations from the fBm model. We also observed that the CI-based estimator,

being non-asymptotic, was considerably conservative.

BIBLIOGRAPHY

Arneodo, A., d’Aubenton Carafa, Y., Bacry, E., Grave, P.V., Muzy, J.F., and Thermes, C.

(1996). Wavelet based fractal analysis of DNA sequences. Physica D: Nonlinear Phenomena,

96, 291–320.

Bardet, J. M., Philippe, A., Oppenheim, G., Taqqu, M. S., Stoev, S. A., and Lang, G. (2003).

Semi-parametric estimation of the long-range dependence parameter: A survey. Theory and

applications of long-range dependence, 2003, 557–577.

Beran, J. (1994). Statistics for long-memory processes. Chapman & Hall/CRC.

Biskamp, D. (1994). Cascade models for magnetohydrodynamic turbulence. Physical Review

E, 50, 2702–2711.

Breton, J.-C. and Coeurjolly, J.-F. (2010). Confidence intervals for the hurst parameter of

a fractional brownian motion based on concentration inequalities. Stat. Infer. Stoch. Proc.,

to appear.

Breton, J.C., Nourdin, I., and Peccati, G. (2009). Exact confidence intervals for the hurst

parameter of a fractional brownian motion. Electron. J. Statist., 3, 416–425.

Coeurjolly, J.-F. (2001). Estimating the parameters of a fractional Brownian motion by

discrete variations of its sample paths. Stat. Infer. Stoch. Proc., 4, 199–227.

Coeurjolly, J.-F. (2008). Hurst exponent estimation of locally self-similar gaussian processes

using sample quantiles. Annals of Statistics, 36, 1404–1434.

Daubechies, I. (2006). Orthonormal bases of compactly supported wavelets. Communica-

tions on Pure and, Applied Mathematics, 41, 909–996.

22

of 26



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

Doukhan, P., Oppenheim, G., and Taqqu, M. S. (2003). Theory and applications of long-

range dependence. Birkhauser.

Horbury, T. S., Forman, M., and Oughton, S. (2008). Anisotropic scaling of magnetohydro-

dynamic turbulence. Phys. Rev. Lett., 101, 175005.

Kolmogorov, A.N. (1941). The local structure of turbulence in incompressible viscous fluid

for very large Reynolds numbers. Dokl. Akad. Nauk SSSR, 30, 9–13.

Mandelbrot, B. and Ness, J. V. (1968). Fractional brownian motions, fractional noises and

applications. SIAM Review, 10, 422–437.

Nelkin, M. (1975). Scaling theory of hydrodynamic turbulence. Phys. Rev. A, 11, 1737–

1743.

Nelkin, M. (1975). Scaling theory of hydrodynamic turbulence. Phys. Rev. A, 11, 1737–

1743.

Nourdin, I. and Reveillac, A. (2009). Asymptotic behavior of weighted quadratic variations

of fractional Brownian motion: the critical case H= 1/4. The Annals of Probability, 37,

2200–2230.

Peligrad, M. and Sethuraman, S. (2008). On fractional brownian motion limits in one dimen-

sional nearest-neighbor symmetric simple exclusion. Latin American Journal of Probability

and Mathematical Statistics, 4, 245–255.

Percival, D. B. and Walden, A. T. (2000). Wavelet Methods for Time Series Analysis.

Cambridge University Press.

Perez, D.G., Zunino, L., and Garavaglia, M. (2004). Modeling turbulent wave-front phase as

a fractional Brownian motion: a new approach. Journal of the Optical Society of America

A, 21, 1962–1969.

Ribak, E. (1997). Atmospheric turbulence, speckle, and adaptive optics. Annals of the New

23

of 26



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

York Academy of Sciences, 808, 193–204.

Schwartz, C., Baum, G., and Ribak, E. (1994). Turbulence-degraded wave fronts as fractal

surfaces. Journal of the Optical Society of America A, 11, 444–451.

Swanson, J. (2010). Fluctuations of the empirical quantiles of independent brownian motions.

Stochastic Processes and their Applications, 121, 479–514.

Veitch, D. and Abry, P. (1999). A wavelet-based joint estimator of the parameters of

long-range dependence. IEEE Transactions on Information Theory, 45, 878–897.

24

of 26



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

variance estimation for fractional brownian … · for peer review only variance estimation for...

Documents