Expectiles for subordinated Gaussian processes with applications
aa r X i v : . [ m a t h . S T ] J u l Expectiles for subordinated Gaussian processeswith applications
Jean-Fran¸cois Coeurjolly and Hedi Kortas Corresponding author:
Laboratory Jean Kuntzmann, Department of Statistics, Grenoble University,121 Avenue Centrale, 38040 Grenoble Cedex 9, France. Higher Institute of Management, Department of Quantitative Methods,Sousse University, Tunisia.September 18, 2018 bstract In this paper, we introduce a new class of estimators of the Hurst exponent of thefractional Brownian motion (fBm) process. These estimators are based on sample expectilesof discrete variations of a sample path of the fBm process. In order to derive the statisticalproperties of the proposed estimators, we establish asymptotic results for sample expectilesof subordinated stationary Gaussian processes with unit variance and correlation functionsatisfying ρ ( i ) ∼ κ | i | − α ( κ ∈ R ) with α >
0. Via a simulation study, we demonstrate therelevance of the expectile-based estimation method and show that the suggested estimatorsare more robust to data rounding than their sample quantile-based counterparts.
Keywords: expectiles; robustness; local shift sensitivity; subordinated Gaussian process;fractional Brownian motion.
In the statistic literature, there has been a tremendous interest in analysis, estimation andsimulation issues pertaining to the fractional Brownian motion (fBm) (Mandelbrot and Ness,1968). This is due to the fact that the fBm process offers an adequate modeling framework fornonstationary self-similar stochastic processes with stationary increments and can be used tomodel stochastic phenomena relating to various fields of research. A fractional Brownian motion(fBm), denoted { B H ( t ) , t ∈ R } with Hurst exponent 0 < H <
1, is a zero-mean continuous-timeGaussian stochastic process whose correlation function satisfies E [ B H ( t ) B H ( s )] = σ ( | t | H + | s | H − | t − s | H ) for all pairs ( t, s ) ∈ R × R and σ = E ( B H (1) ). The fBm is H -self-similari.e., for all α > , B H ( αt ) d = α H B H ( t ), where d = means the equality of all its finite-dimensionalprobability distributions. The process corresponding to the first-order increments of the fBm isknown as the fractional Gaussian noise (fGn) whose correlation function ρ H ( i ) is asymptoticallyof the order of | i | H − for large lag lengths i . In particular, for 1 / < H <
1, the correlations arenot summable, i.e. P + ∞ i = −∞ ρ H ( i ) = ∞ . This property is referred to as long-range dependenceor long-memory whereas the case 0 < H < / ρ ( i ) ∼ κ | i | − α ( κ ∈ R ) with α >
0. A short sim-ulation study is conducted to corroborate our theoretical findings. In Section 3, we discuss asample expectile-based estimator of the Hurst exponent and derive its statistical properties. Wethen perform a simulation study in order to confirm the effectiveness of the suggested estimationmethod. 3
Expectiles for subordinated Gaussian processes
Given some random variable Z with mean µ , F Z is referred to the cumulative distributionfunction of Z and ξ Z ( p ) for p ∈ (0 ,
1) to its p th quantile. It is well-known that the p th quantile ofa random variable Z can be obtained by minimizing asymmetrically the weighted mean absolutedeviation ξ Z ( p ) := argmin θ E (cid:2) | p − Z ≤ θ | . | Z − θ | (cid:3) . In order to limit the local shift sensitivity of the p th quantile, Newey and Powell (1987) definedthe notion of expectile denoted by E Z ( p ) for some p ∈ (0 , E Z ( p ) := argmin θ E (cid:2) | p − Z ≤ θ | . ( Z − θ ) (cid:3) . (1)We may note that the 50%-expectile if nothing else than the expectation of Z . Newey and Powell(1987) argued that providing E [ Z ] < + ∞ , then for every p ∈ (0 ,
1) the solution of (1) is uniqueon the set I F Z := { x ∈ R : F Z ( x ) ∈ (0 , } . The expectile can also be defined as the solution ofthe equation E (cid:2) | p − Z ≤ θ | . ( Z − θ ) (cid:3) = 0.A key property of the expectile is that it is scale and location equivariant (Newey and Powell,1987). The scale equivariance property means that for Y = aZ where a >
0, the p th expectile of Y satisfies: E Y ( p ) = aE Z ( p ) (2)The p th expectile is location equivariant in the sense that for Y = Z + b where b ∈ R , the p th expectile of Y is such that: E Y ( p ) = E Z ( p ) + b (3)Now, let Z = ( Z , . . . , Z n ) be a sample of identically distributed random variables withcommon distribution F Z , the sample expectile of order p is defined as: b E ( p ; Z ) := argmin θ n n X i =1 | p − Z i ≤ θ | . ( Z i − θ ) . In order to derive asymptotic results for Hurst exponent estimates based on expectiles, we haveto provide asymptotic results for sample expectiles of nonlinear functions of (centered) subor-dinated stationary Gaussian processes with variance 1 and with correlation function decreasing4yperbolically. This will be the setting of the rest of this section. Let { Y i } + ∞ i =1 be such a Gaus-sian process with correlation function ρ ( · ) satisfying ρ ( i ) ∼ κ | i | − α for κ ∈ R and α >
0. Let Y = ( Y , . . . , Y n ) a sample of n observations and h ( Y ) = ( h ( Y ) , . . . , h ( Y n )) its subordinatedversion for some measurable function h . We wish to provide asymptotic results for the sample p th expectile defined by b E ( p ; h ( Y )) := argmin θ n n X i =1 (cid:12)(cid:12) p − h ( Y i ) ≤ θ (cid:12)(cid:12) . ( h ( Y i ) − θ ) . (4)Since the criterion is differentiable in θ , the sample p th expectile also satisfies the followingestimating equation ψ n (cid:16) b E ( p ; h ( Y )) ; h ( Y ) (cid:17) = 0 with ψ n ( θ ; h ( Y )) := 1 n n X i =1 (cid:12)(cid:12) p − h ( Y i ) ≤ θ (cid:12)(cid:12) . ( h ( Y i ) − θ ) . (5)In the following, we need the two following additional notation for Y ∼ N (0 , ψ h ( Y ) ( θ ; p ) := E (cid:2)(cid:12)(cid:12) p − h ( Y ) ≤ θ (cid:12)(cid:12) . ( h ( Y ) − θ ) (cid:3) ψ ′ h ( Y ) ( θ ; p ) := − E (cid:2)(cid:12)(cid:12) p − h ( Y ) ≤ θ (cid:12)(cid:12)(cid:3) = − p (1 − F h ( Y ) ( θ )) − (1 − p ) F h ( Y ) ( θ ) , the latter quantity corresponding to the derivative of ψ h ( Y ) ( · , p ) if it is well-defined. Let us notethat the p th expectile of h ( Y ) satisfies ψ h ( Y ) ( E h ( Y ) ; p ) = 0. We now present the assumption onthe function h considered in our asymptotic result. [A(h,p)] h ( · ) is a measurable function such that E h ( Y ) < + ∞ and such that the function ψ h ( Y ) ( · , p ) is continuously differentiable in a neighborhood of E h ( Y ) ( p ) with negative derivativeat this point.Such an assumption is in particular satisfied under the following one: [A ′ ( h ) ] h ( · ) is a measurable function such that E h ( Y ) < + ∞ , h is not “flat”, i.e. for all θ ∈ R the set { y ∈ R : h ( y ) = θ } has null Lebesgue measure.Indeed, if h satisfies [A ′ ( h ) ] then ψ ( · , p ) is differentiable in θ . And since, E h ( Y ) ( p ) belongsto the set I h ( Y ) = { x ∈ R : F h ( Y ) ( x ) ∈ (0 , } , ψ ′ ( E h ( Y ) ( p ); p ) is necessarily negative. Forthe purpose of this paper, our main result will be applied with h ( · ) = | · | β (with β >
0) or h ( · ) = log | · | which obviously satisfy [A ′ ( h ) ] . 5he nature of the asymptotic result will depend on the correlation structure of the Gaussianprocess and on the Hermite rank, τ ( p, θ ) of the function e ψ ( t ; p, θ ) := (cid:12)(cid:12) p − h ( t ) ≤ θ (cid:12)(cid:12) . ( h ( t ) − θ ) − ψ h ( Y ) ( θ ; p ) . We recall that the Hermite rank (see e.g.
Taqqu (1977)) corresponds to the smallest integersuch that the coefficient in the Hermite expansion of the considered function is not zero. For thesake of simplicity, assume that the Hermite rank of this function depends neither on θ nor p anddenote it simply by τ . Again, this could be weakened since we believe that the next result couldbe proved with the following Hermit rank: inf θ ∈V ( E h ( Y ); p ) τ ( p, θ ). As an example, the Hermiterank of e ψ ( · , p, θ ) is 1 for h ( · ) = · and ( p, θ ) ∈ (0 , × R and 2 for h ( · ) = | · | β ( β >
0) or log | · | for( p, θ ) ∈ (0 , × R + \ { } . We now present our main result stating a Bahadur type representationfor the sample p th expectile of a subordinated Gaussian process. Theorem 1
Let { Y i } + ∞ i =1 a (centered) stationary Gaussian process with variance 1 and correla-tion function satisfying ρ ( i ) ∼ κ | i | − α ( κ ∈ R ), as | i | → + ∞ with α > and with a function h satisfying [A(h,p)] . Let h ( Y ) = ( h ( Y ) , . . . , h ( Y n )) a sample of n observations of the subordi-nated process, then, for all p ∈ (0 , b E ( p ; h ( Y )) − E h ( Y ) ( p ) = − ψ n (cid:0) E h ( Y ) ( p ); h ( Y ) (cid:1) ψ ′ (cid:0) E h ( Y ) ( p ); p (cid:1) + o P ( r n ) , (6) where the sequence r n = r n ( α, τ ) is defined by r n = n − / if ατ > n − / log( n ) if ατ = 1 n − ατ/ if ατ > . Proof.
Let us simplify the notation for sake of conciseness: let b E = b E ( p ; h ( Y )), E = E h ( Y ) ( p ), ψ n ( E ) = ψ n ( E h ( Y ) ( p ); h ( Y )) and ψ ′ ( E ) = ψ ′ h ( Y ) ( E h ( Y ) ( p ); p ). The first thing to noteis that the sequence r n corresponds to the short-range or long-range dependence characteristic ofthe sequence e ψ ( h ( Y ); p, θ ) , . . . , e ψ ( h ( Y n ); p, θ ). More precisely r n corresponds to the asymptoticbehavior of E ψ n ( E ) . Indeed, if ( c j ) j ≥ denotes the sequence of the Hermite coefficients of theexpansion of e ψ ( · ; p, E ) in Hermite polynomials (denoted by ( H j ( t )) j ≥ and normalized in sucha way that E [ H j ( Y ) H k ( Y )] = j ! δ jk ), we may have using standard developments on Hermite6olynomials (see e.g. Taqqu (1977)) E ψ n ( E ) = 1 n n X i,j =1 E h e ψ ( Y i ; p, E ) e ψ ( Y j ; p, E ) i = 1 n n X i,j =1 X k ,k ≥ c k c k k ! k ! E [ H k ( Y i ) H k ( Y j )]= 1 n n X i,j =1 X k ≥ τ c k k ! ρ ( j − i ) k = O (cid:16) n X | i |≤ n | ρ ( i ) | τ | {z } =: ρ n (cid:17) = O ( r n ) . (7)Let us define V n := r − n ( b E − E ) and W n ( E ) := − r − n ψ n ( E ) /ψ ′ ( E ). We just have to provethat V n − W n ( E ) converges in probability to 0 as n → + ∞ . The proof is based on the applicationof Lemma 1 of Ghosh (1971) which consists in satisfying the two following conditions:( a ) for all δ >
0, there exists ε = ε ( δ ) such that P ( | W n ( E ) | > ε ) < δ .( b ) for all y ∈ R and for all ε > n → + ∞ P ( V n ≤ y, W n ( E ) ≥ y + ε ) = lim n → + ∞ P ( V n ≥ y + ε, W n ( E ) ≤ y ) = 0 . ( a ) is in particular fulfilled if we prove that E W n ( E ) = O (1) which follows from (7) since E W n ( E ) = ψ ′ ( E ) − r − n E ψ n ( E ) = r − n × O ( ρ n ) = O (1).( b ) We consider only the first limit. The second one follows similar developments. We firststate that the map ψ n ( · ) is decreasing. Indeed, let θ ≤ θ ′ and denote by Z i ( θ ) the variable | p − h ( y i ) ≤ θ ) | . ( h ( Y i ) − θ ). If h ( Y i ) ≤ θ or h ( Y i ) > θ , we obviously get Z i ( θ ) > Z i ( θ ′ ) a.s. leadingto the decreasing of ψ n ( · ). And in the in between case, Z i ( θ ) − Z i ( θ ′ ) = p ( θ ′ − θ ) + θ ′ − h ( Y i ) ≥ y ∈ R , then also using the fact that ψ n ( b E ) = ψ ( E ) = 0and ψ ′ ( E ) <
0, we derive { V n ≤ y } = { b E ≤ y × r n + E } = { ψ n ( b E ) ≥ ψ n ( y × r n + E ) } = { ψ ( y × r n + E ) − ψ n ( y × r n + E ) ≥ ψ ( y × r n + E ) − ψ ( E ) } = { W n ( y × r n + E ) ≤ y n } , where y n = r − n ψ ′ ( E ) − ( ψ ( y × r n + E ) − ψ ( E )). Under the assumption [A(h,p)] , y n → y as n → + ∞ . Now, let U n := ψ ′ ( E ) ( W n ( E ) − W n ( y × r n + E )), explicitly given by U n = 1 nr n n X i =1 (cid:16) e ψ ( Y i ; p, E + y × r n ) − e ψ ( Y i ; p, E ) (cid:17) . c j,n the j th Hermite coefficient of the function r − n (cid:16) e ψ ( t ; p, E + y × r n ) − e ψ ( t ; p, E ) (cid:17) , thenunder the assumption [A(h,p)] and from the dominated convergence theorem we can prove that c j,n n → + ∞ −→ y E (cid:2) | p − h ( Y ) ≤ E | H j ( Y ) (cid:3) =: e c j . Therefore for n large enough, E [ U n ] = 1 n n X i,j =1 X k ≥ τ c k,n k ! ρ ( j − i ) k ≤ n X | i |≤ n X k ≥ τ e c k k ! ρ ( i ) τ = O ( ρ n ) = O ( r n )which leads to the convergence of U n to 0 in probability. For all ε >
0, there exists n ( ε ) suchthat for all n ≥ n ( ε ), y n ≤ y + ε/
2. Therefore for n ≥ n ( ε ) P ( V n ≤ y , W n ≥ y + ε ) = P ( W n ( y × r n + E ) ≤ y n , W n ≥ y + ε ) ≤ P ( W n ( y × r n + E ) ≤ y + ε/ , W n ( E ) ≥ y + ε ) ≤ P ( | W n ( y × r n + E ) − W n ( E ) | ≥ ε/ n → + ∞ → , which ends the proof.In the case of short-range dependence, i.e. ατ > Corollary 2 ( i ) Under the assumptions of Theorem 1 with p ∈ (0 , and ατ > , then as n → + ∞√ n (cid:16) b E ( p ; h ( Y )) − E h ( Y ) ( p ) (cid:17) d −→ N (0 , σ ( p )) , where σ ( p ) = 1 ψ ′ (cid:0) E h ( Y ) ( p ); p (cid:1) X i ∈ Z X k ≥ τ c k ( p ) k ! ρ ( i ) k and where c k ( p ) is the k th Hermite coefficient of the expansion of the function ψ ( h ( · ); E h ( Y ) ( p ); p ) in Hermite polynomials. ii ) Let { Y i } + ∞ i =1 and { Y i } + ∞ i =1 two (centered) stationary Gaussian processes with variances 1and correlation functions (resp. cross-correlation functions) ρ , ρ (resp. ρ , ρ ) decreasinghyperbolically with exponents α , α (resp. α , α ). Let p ∈ (0 , , h a function satisfying [A(h,p)] and let h ( Y ) and h ( Y ) be the samples of n observations of the two subordinatedsamples. If min( α , α , α , α ) × τ > , then as n → + ∞√ n (cid:16) b E (cid:0) p ; h ( Y ) (cid:1) − E h ( Y ) ( p ) , b E (cid:0) p ; h ( Y ) (cid:1) − E h ( Y ) ( p ) (cid:17) T d −→ N (0 , Σ ) . where Σ is the (2 , matrix with entries Σ ab for a, b = 1 , given by Σ ab = 1 ψ ′ (cid:0) E h ( Y ) ( p ); p (cid:1) X i ∈ Z X k ≥ τ c k ( p ) k ! ρ ab ( i ) k . (8)As it was established for sample quantiles (Coeurjolly, 2008), a non standard limit towards aRosenblatt process is expected in the other cases ( ατ ≤ To illustrate a part of the previous results, we propose a short simulation study in this section.The latent stationary Gaussian process we consider here is the fractional Gaussian noise withvariance 1, which is obtained by taking the discretized increments from a fractional Brownianmotion. The correlation function of the fractional Gaussian noise with Hurst parameter (orself-similarity parameter) H ∈ (0 ,
1) satisfies the hyperbolic decreasing property required inTheorem 1 with α = 2 − H . Discretized sample paths of fractional Brownian motion can begenerated exactly using the embedding circulant matrix method popularized by Wood and Chan(1994) (see also Coeurjolly (2000)) which is implemented in the R package dvfBm .Figures 1 and 2 illustrate the convergence of the sample expectiles. Three h functions areconsidered: h ( · ) = ( · ), ( · ) and log | · | . The related Hermite rank of the function e ψ is respectively1,2 and 2 for these three h functions. The sample size of the simulation is fixed to n = 500. Wecan claim the convergence of the sample expectile b E ( p ; h ( Y )) towards E h ( Y ) ( p ) for all the valuesof α (or H ), p and for the three functions h considered. If we focus on h ( · ) = ( · ), we can alsoremark a higher variance of the sample estimates for α = 0 . α = 1 .
4. This is inagreement with the theory since for α = 0 . ατ = 0 . < n − / which means an increasing of the variance. For the two other functions considered,then ατ is always greater than 1 (it equals either 2.8 or 1.2 in our simulations) and we do notobserve such an increasing of the variance. 9o put emphasis on this last point, Figure 3 shows in log-scale the average (over the 9 orderof expectiles considered in the simulation, i.e. p = 0 . , . . . , .
9) of the empirical variances interms of n for the three h functions and for the two values of α = 0 . α = 1 .
4. We clearlyobserve that as soon as ατ >
1, the slope of the curves is close to − h ( · ) = 1 and α = 0 .
6, we observe that theslope is about − . n − ατ which is expected fromTheorem 1.Figure 1: Boxplots of sample expectiles for expectiles of order p = 0 . , . . . , . m = 500replications of fractional Gaussian noise with length n = 500 and with Hurst parmeter H = 0 . α = 1 .
4) and H = 0 . α = 0 . h functions considered here is the identityfunction (with Hermite rank 1). The curves correspond to the theoretical expectile functions for Y ∼ N (0 , Let X = ( X ( i )) i =1 ,...,n be a discretized version of a fractional Brownian motion process and let a be a filter of length ℓ + 1 and of order ν ≥ ℓ X q =0 q j a q = 0 , for j = 0 , . . . , ν − ℓ X q =0 q ν a q = 0 . Define also X a to be the series obtained by filtering X with a , then: X a ( i ) = ℓ X q =0 a q X ( i − q ) , for i ≥ ℓ + 110igure 2: Boxplots of sample expectiles for expectiles of order p = 0 . , . . . , . m = 500replications of fractional Gaussian noise with length n = 500 and with Hurst parmeter H = 0 . α = 1 .
4) and H = 0 . α = 0 . h functions with Hermite rank 2 have beenconsidered here: h ( · ) = ( · ) (top) and h ( · ) = log | · | (bottom). The curves correspond to thetheoretical expectile functions for Y (middle) and log | Y | (bottom) where Y ∼ N (0 , n in log-scale based on m = 500 replications of fractional Gaussian noise with parameters H = 0 . α = 1 .
4) and H = 0 . α = 0 . . , . . . , .
9) forthe orders of the expectiles and we compute b σ n = 1 / × P i =1 b σ i,n where b σ i,n is the empiricalvariance for the expectile with order i/
10 for the sample size n . Three choices of h functionshave been considered: h ( · ) = ( · ) , ( · ) and log | · | .11nd ˜X a as the normalized vector of X a , i.e.: ˜X a = X a E (( X a (1)) ) / . It should be noticed here that the filtering operation allows to decorrelate the increments of thediscretized version of the fractional Brownian motion process. Indeed, it may be proved (see e.g.
Coeurjolly (2001)) that: ρ aH ( i ) ∼ k H | i | H − ν as | i | → + ∞ .Consider the sequence ( a m ) m ≥ defined by: a mi = a j if i = jm i = 0 , . . . , mℓ , which is the filter a dilated m times. It has been shown in Coeurjolly (2001, 2008) that:˜ X a m = X a m σ m where σ m = m H σ κ a H and κ a H = − P ℓq,q ′ =0 a q a q ′ | q − q ′ | H .The following proposition allows us to construct an ordinary least squares (OLS) estimatorof the Hurst exponent H of a fBm process based on sample expectiles. Proposition 3
Let b E (cid:0) p ; h ( X a m ) (cid:1) and b E (cid:16) p ; h ( ˜X a m ) (cid:17) be the p th order sample expectiles for thefiltered series h ( X a m ) and h ( ˜X a m ) respectively. Here two positive functions h ( · ) are considered,namely: h ( · ) = | · | β for β > and h ( · ) = log | · | . We have: b E (cid:16) p ; | X a m | β (cid:17) = σ βm b E (cid:16) p ; | ˜ X a m | β ) (cid:17) (9) and b E (cid:16) p ; log | X a m | (cid:17) = 12 log( σ m ) + b E (cid:16) p ; log | ˜X a m | (cid:17) . (10) Proof.
We have: b E (cid:16) p ; | X a m | β (cid:17) = argmin θ n − mℓ n − X i = mℓ | p − {| X a m ( i ) | β ≤ θ } | . ( | X a m ( i ) | β − θ ) = argmin θ n − ml n − X i = ml | p − {| ˜ X a m ( i ) | β ≤ θσβm } | . ( | ˜ X a m ( i ) | β − θσ βm ) . Setting θ ′ = θσ βm , the proof of the first relation (9) follows easily. Using the same methodology,we can demonstrate the result given by equation (10).12 emark 1 It should be stressed here that the scaling relationship relating the theoretical p thexpectiles for the series h ( X a m ) and h ( ˜X a m ) can be obtained directly using the scale equivarianceproperty (2) for h ( · ) = | · | β and the location equivariance property (3) for h ( · ) = log | · | . Now applying the logarithmic transformation to both sides of (9), we get:log b E (cid:16) p ; | X a m | β (cid:17) = βH log( m ) + log (cid:16) σ β ( κ a H ) β/ E | Y | β ( p ) (cid:17) + log b E (cid:16) p ; | ˜X a m | β (cid:17) E | Y | β ( p ) . (11)On the other hand, (10) can be reformulated in the following way: b E (cid:16) p ; log | X a m | (cid:17) = H log( m ) + 12 log( σ κ a H ) + E log | Y | ( p ) + (cid:16) b E (cid:16) p ; log( | ˜X a m | ) (cid:17) − E log | Y | ( p )) (cid:17) . (12)It is noteworthy here that we expect that log b E (cid:16) p ; | ˜X a m | β (cid:17) /E | Y | β ( p ) and b E (cid:16) p ; log( | ˜X a m | β ) (cid:17) − E log | Y | ( p ) to converge towards 0 as n → ∞ . Hence, based on equations (11) and (12), we opt foran OLS regression scheme. This allows to derive the two following estimators of the hurst indexdefined by: b H β = A T β || A || (cid:16) log b E (cid:16) p ; | X a m | β (cid:17)(cid:17) m =1 ,...,M , (13)and b H log = A T || A || (cid:16) b E (cid:16) p ; log | X a m | (cid:17)(cid:17) m =1 ,...,M , (14)where A is the vector of length M with components A m = log m − M P Mm =1 log( m ), m = 1 , . . . , M for some M ≥ || z || for some vector z of length d designates the norm defined by (cid:16)P di =1 z i (cid:17) / . Notice here that b H β and b H log do not depend on σ .We would like to put the stress on the fact that (13) and (14) are really similar to the onesdeveloped in Coeurjolly (2001, 2008). Indeed, the standard procedure developed in Coeurjolly(2001) simply consists in replacing the sample expectile by the sample variance (this method willbe denoted by ST in Section 3.2). To deal with outliers, the procedure developed in Coeurjolly(2008) consists in replacing the sample expectile by either the sample median of ( X a m ) or thetrimmed-means of ( X a m ) . These two last methods are denoted by MED and TM in Section 3.2.Now, we state the asymptotic results for these new estimates based on expectiles. Proposition 4
Let a a filter with order ν ≥ , p ∈ (0 , , β > then as n → + ∞ , b H β and b H log converge in probability to H . Moreover, the following convergences in distribution hold √ n (cid:16) b H β − H (cid:17) d −→ N (0 , σ β ) and √ n (cid:16) b H log − H (cid:17) d −→ N (0 , σ ) , here σ β = 1 E | Y | β ( p ) × A T Σ β A β k A k and σ = A T Σ log A k A k and where the ( M, M ) matrices Σ β and Σ log are defined by (8). Proof.
We only provide a sketch of the proof. We claim that once Theorem 1 and Corollary 2are established, the obtention of convergences stated in Proposition 4 are semi-routine. First ofall, let us notice that b H β − H = A T β k A k log b E (cid:16) p ; | e X a m | β (cid:17) E | Y | β m =1 ,...,M (15)and b H log − H = A T β k A k (cid:16) b E (cid:16) p ; log | e X a m | (cid:17) − E log | Y | ( p ) (cid:17) m =1 ,...,M . (16)Since the functions | · | β and log | · | have Hermite rank 2 and since the correlation function ofthe stationary sequence e X a m decreases hyperbolically with an exponent α = 2 ν − H then forany m ∈ { , . . . , M } , Theorem 1 holds with r n = n − / (since ατ > H ∈ (0 , e X a m and e X a m is defined by ρ a m , a m H ( j ) = π a m , a m H ( j ) π a m , a m H (0) / π a m , a m H (0) / with π a m , a m H ( j ) = ℓ X q,r =0 a q a r | m q − m r + j | H . Lemma 1 in Coeurjolly (2008) states that for all m , m the correlation function ρ a m , a m H is alsodecreasing hyperbolically with an exponent α = 2 ν − H , then Corollary 2 may be applied toprove that (cid:16) b E (cid:16) p ; | e X a m | β (cid:17) − E | Y | β (cid:17) m =1 ,...,M d −→ N (0 , Σ β ) (17)and (cid:16) b E (cid:16) p ; log | e X a m | (cid:17) − E log | Y | (cid:17) m =1 ,...,M d −→ N (0 , Σ log ) , (18)where according to (8), the ( M, M ) matrices Σ β and Σ log are respectively defined byΣ βm m = 1 ψ ′ (cid:0) E | Y | β ( p ); p (cid:1) X i ∈ Z X k ≥ c βk ( p ) k ! ρ a m , a m H ( i ) k (19)Σ log m m = 1 ψ ′ (cid:0) E log | Y | ( p ); p (cid:1) X i ∈ Z X k ≥ c log k ( p ) k ! ρ a m , a m H ( i ) k , (20)14here ( c βk ) k ≥ and ( c log k ) k ≥ are respectively the Hermite coefficients of the functions | · | β andlog | · | . The convergences (17) and (18) combined with (15) and (16) and the use of the delta-method (for the convergence of b H β ) end the proof. In this section, we investigate the interest of the new estimators based on expectiles. We considerthree different models in our simulations.(a) standard fBm : non-contaminated fractional Brownian motion.(b) fBm with additive outliers : we contaminate 5% of the observations of the increments of thefractional Brownian motion with an independent Gaussian noise such that the SNR of theconsidered components equals − Db .(c) rounded fBm : we assume the data are given by the integer part of a discretized samplepath of an original fBm.To fix ideas, Figure 4 provides some examples of discretized sample paths of standard andcontaminated fBm. The simulation results are presented in Tables 1 and 2. For these simulations,as suggested in Coeurjolly (2001), we chose the filter a = d M = 5. Also, in other simulations not presented here, we have observed that the estimates b H β perform better than b H log and, among all possible choices of β , the value β = 2 seems to bea good compromise. Therefore, we present only the result for this latter estimator, that is b H β with β = 2.In a first step, we had observed a quite large sensitivity to the value of the probability p defining the expectile. In order to have an efficient data-driven procedure, we propose to choosethe probability parameter p via a Monte-Carlo approach as follows:1. Estimate the parameters H and σ using the standard method (the estimation of σ isnot described here but it may be found for example in Coeurjolly (2001)). Denote theseestimates b H and b σ .2. Generate B = 100 contaminated fBm with Hurst parameter b H and scaling coefficient b σ ,define a grid of probabilities ( p , . . . , p P ). For each new replication, we estimate b H withexpectiles for all the p i . The optimal p , denoted in the tables by p opt , is then defined asthe one achieving the smallest mean squared error (based on the B = 100 replications).15he procedure based on expectiles, denoted E(p) in the results, is compared to the standardmethod (ST) and to methods which efficiently deal with outliers, that is methods MED and TM(the last one is calculated by discarding 5% of the lowest and the highest values of ( X a m ) ateach scale m ).The standard fBm model is used as a control to show that all methods perform well. Asseen in the first two columns of Tables 1 and 2, this is indeed true. All the methods seem to beasymptotically unbiased and have a variance converging to zero. We can also remark that in thissituation whatever the value of H , estimates based on expectiles exhibit a performance which isvery close to the one of the standard method (wich can also be viewed as the method based onexpectile with p = 0 . rounded fBm corresponding to the last two columns of each table.In this situation, expectiles are shown to be more efficient in terms of bias and its variance seemsto be not too much affected by this type of strong contamination. We also put the stress on theinterest and efficiency to choose the p value based on a Monte-Carlo approach. References
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