Expectiles for subordinated Gaussian processes with applications
In this paper, we introduce a new class of estimators of the Hurst exponent of the fractional Brownian motion (fBm) process. These estimators are based on sample expectiles of discrete variations of a sample path of the fBm process. In order to derive the statistical properties of the proposed estimators, we establish asymptotic results for sample expectiles of subordinated stationary Gaussian processes with unit variance and correlation function satisfying () with . Via a simulation study, we demonstrate the relevance of the expectile-based estimation method and show that the suggested estimators are 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 and simulation 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 for nonstationary self-similar stochastic processes with stationary increments and can be used to model stochastic phenomena relating to various fields of research. A fractional Brownian motion (fBm), denoted with Hurst exponent , is a zero-mean continuous-time Gaussian stochastic process whose correlation function satisfies for all pairs and . The fBm is -self-similar i.e., for all , where means the equality of all its finite-dimensional probability distributions. The process corresponding to the first-order increments of the fBm is known as the fractional Gaussian noise (fGn) whose correlation function is asymptotically of the order of for large lag lengths . In particular, for , the correlations are not summable, i.e. . This property is referred to as long-range dependence or long-memory whereas the case corresponds to short memory.
Several methods aimed at estimating the Hurst characteristic exponent or long-memory exponent have been developed. Among these statistical methods figure the Fourier-based methods such as the Whittle maximum likelihood estimator (see e.g. Beran (1994); Robinson (1995)) or the spectral regression based estimator (Beran, 1994). The wavelet estimators have been also extensively investigated either with an ordinary least squares (Flandrin, 1992), a weighted least squares (see e.g. Abry et al. (2000, 2003); Bardet et al. (2000); Soltani et al. (2004)) or a maximum likelihood (see e.g. (Wornell and Oppenheim, 1992; Percival and Walden, 2000)) estimation schemes. Faÿ et al. (2009) present a deep analysis of Fourier and wavelet methods. Recently, the so-called discrete variations techniques (see e.g.Kent and Wood (1997); Istas and Lang (1997); Coeurjolly (2001)) have been introduced. Within this class of estimators, Coeurjolly (2008) proposed a new method based on sample quantiles to estimate the Hurst exponent in the more general setting of locally self-similar gaussian processes. This estimator has been proven robust when dealing with outliers (Achard and Coeurjolly, 2010). The latter, often encountered in real world applications, can induce a significant estimation bias. Actually, the advantage of quantiles is that they have a bounded gross-error-sensitivity (Hampel et al., 1986; Huber, 1981) allowing them to cope efficiently with the problem of outliers. Nevertheless, this is not the only problem faced when dealing with estimation issues. Indeed, data rounding is also a serious impediment. Data rounding is common in finance (Bijwaard and Franses, 2009; Rosenbaum, 2009), economics (Williams, 2006), computer science (Matthieu, 2006; Bois and Vignes, 1982) and computational physics (Vilmart, 2008) and can lead to several misinterpretations. Quantiles are unfortunately not robust against data rounding since their local shift sensitivity is unbounded, see Hampel et al. (1986); Huber (1981). Newey and Powell (1987) have introduced the so-called expectile which, although similar to quantile, has a bounded local shift-sensitivity and thus can handle the rounding issue.
In this paper, we derive a Bahadur-type representation for sample expectiles of a subordinated Gaussian process with unit variance and correlation function with hyperbolic decay. This allows us to investigate the statistical properties of a new discrete variations estimator of the Hurst exponent of the fBm process. In constructing this estimator, we rely mainly on the scale and location equivariance properties of expectiles (Newey and Powell, 1987). We will show via a simulation study the robustness of the proposed estimator against data rounding.
The remainder of this paper is structured as follows: Section 2 deals with asymptotic properties of sample expectiles for a class of subordinated stationary Gaussian processes with unit variance and correlation function satisfying () with . A short simulation study is conducted to corroborate our theoretical findings. In Section 3, we discuss a sample expectile-based estimator of the Hurst exponent and derive its statistical properties. We then perform a simulation study in order to confirm the effectiveness of the suggested estimation method.
2 Expectiles for subordinated Gaussian processes
2.1 A few notation
Given some random variable with mean , is referred to the cumulative distribution function of and for to its th quantile. It is well-known that the th quantile of a random variable can be obtained by minimizing asymmetrically the weighted mean absolute deviation
In order to limit the local shift sensitivity of the th quantile, Newey and Powell (1987) defined the notion of expectile denoted by for some . Rather than an absolute deviation (function), a quadratic loss function is considered:
We may note that the -expectile if nothing else than the
expectation of . Newey and Powell (1987) argued that
providing , then for every the
solution of (1) is unique on the set .
The expectile can also be defined as the solution of the equation .
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 where , the th expectile of satisfies:
The th expectile is location equivariant in the sense that for where , the th expectile of is such that:
Now, let be a sample of identically distributed random variables with common distribution , the sample expectile of order is defined as:
2.2 Main result
In order to derive asymptotic results for Hurst exponent estimates based on expectiles, we have to provide asymptotic results for sample expectiles of nonlinear functions of (centered) subordinated stationary Gaussian processes with variance 1 and with correlation function decreasing hyperbolically. This will be the setting of the rest of this section. Let be such a Gaussian process with correlation function satisfying for and . Let a sample of observations and its subordinated version for some measurable function . We wish to provide asymptotic results for the sample th expectile defined by
Since the criterion is differentiable in , the sample th expectile also satisfies the following estimating equation with
In the following, we need the two following additional notation for
the latter quantity corresponding to the derivative of
if it is well-defined. Let us note that the
th expectile of satisfies .
We now present the assumption on the function considered in our asymptotic result.
[A(h,p)] is a measurable function such that and such that the function is continuously differentiable in a neighborhood of with negative derivative at this point.
Such an assumption is in particular satisfied under the following one:
 is a measurable function such that , is not “flat”, i.e. for all the set has null Lebesgue measure.
Indeed, if satisfies  then is differentiable in . And since, belongs to the set , is necessarily negative. For the purpose of this paper, our main result will be applied with (with ) or which obviously satisfy .
The nature of the asymptotic result will depend on the correlation structure of the Gaussian process and on the Hermite rank, of the function
We recall that the Hermite rank (see e.g. Taqqu (1977)) corresponds to the smallest integer such that the coefficient in the Hermite expansion of the considered function is not zero. For the sake of simplicity, assume that the Hermite rank of this function depends neither on nor and denote it simply by . Again, this could be weakened since we believe that the next result could be proved with the following Hermit rank: . As an example, the Hermite rank of is for and and for () or for . We now present our main result stating a Bahadur type representation for the sample th expectile of a subordinated Gaussian process.
Let a (centered) stationary Gaussian process with variance 1 and correlation function satisfying (), as with and with a function satisfying [A(h,p)]. Let a sample of observations of the subordinated process, then, for all
where the sequence is defined by
Proof. Let us simplify the notation for sake of conciseness: let , , and . The first thing to note is that the sequence corresponds to the short-range or long-range dependence characteristic of the sequence . More precisely corresponds to the asymptotic behavior of . Indeed, if denotes the sequence of the Hermite coefficients of the expansion of in Hermite polynomials (denoted by and normalized in such a way that ), we may have using standard developments on Hermite polynomials (see e.g. Taqqu (1977))
Let us define and . We just have to prove that converges in probability to 0 as . The proof is based on the application of Lemma 1 of Ghosh (1971) which consists in satisfying the two following conditions:
for all , there exists such that .
for all and for all
is in particular fulfilled if we prove that which follows from (7) since .
We consider only the first limit. The second one follows similar developments. We first state that the map is decreasing. Indeed, let and denote by the variable . If or , we obviously get a.s. leading to the decreasing of . And in the in between case, which leads to the same conclusion. Let , then also using the fact that and , we derive
where . Under the assumption [A(h,p)], as . Now, let , explicitly given by
Let the th Hermite coefficient of the function , then under the assumption [A(h,p)] and from the dominated convergence theorem we can prove that
Therefore for large enough,
which leads to the convergence of to 0 in probability. For all , there exists such that for all , . Therefore for
which ends the proof.
In the case of short-range dependence, i.e. then, using the Bahadur type representation of expectiles, we derive immediately the following asymptotic normality for the sample expectile and some generalisations. This result is based on standard central limit theorem for means of subordinated Gaussian stationary processes (Taqqu, 1977; Arcones, 1994).Therefore the proof is omitted.
Under the assumptions of Theorem 1 with and , then as
and where is the th Hermite coefficient of the expansion of the function in Hermite polynomials.
Let and two (centered) stationary Gaussian processes with variances 1 and correlation functions (resp. cross-correlation functions) (resp. ) decreasing hyperbolically with exponents (resp. ). Let , a function satisfying [A(h,p)] and let and be the samples of observations of the two subordinated samples. If , then as
where is the matrix with entries for given by
As it was established for sample quantiles (Coeurjolly, 2008), a non standard limit towards a Rosenblatt process is expected in the other cases (). This case is not considered here.
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 with variance 1, which is obtained by taking the discretized increments from a fractional Brownian motion. The correlation function of the fractional Gaussian noise with Hurst parameter (or self-similarity parameter) satisfies the hyperbolic decreasing property required in Theorem 1 with . Discretized sample paths of fractional Brownian motion can be generated 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 functions are considered: , and . The related Hermite rank of the function is respectively 1,2 and 2 for these three functions. The sample size of the simulation is fixed to . We can claim the convergence of the sample expectile towards for all the values of (or ), p and for the three functions considered. If we focus on , we can also remark a higher variance of the sample estimates for compared to . This is in agreement with the theory since for , and the rate of convergence is lower than which means an increasing of the variance. For the two other functions considered, then is always greater than 1 (it equals either 2.8 or 1.2 in our simulations) and we do not observe such an increasing of the variance.
To put emphasis on this last point, Figure 3 shows in log-scale the average (over the 9 order of expectiles considered in the simulation, i.e. ) of the empirical variances in terms of for the three functions and for the two values of and . We clearly observe that as soon as , the slope of the curves is close to which agrees with the result presented in Corollary 2 for example. When and , we observe that the slope is about which seems to agree with the convergence in which is expected from Theorem 1.
3 Estimation of the Hurst exponent using sample expectiles and discrete variations
3.1 Estimation method and asymptotic results
Let be a discretized version of a fractional Brownian motion process and let be a filter of length and of order with real components i.e.:
Define also to be the series obtained by filtering with , then:
and as the normalized vector of , i.e.:
It should be noticed here that the filtering operation allows to decorrelate the increments of the discretized version of the fractional Brownian motion process. Indeed, it may be proved (see e.g. Coeurjolly (2001)) that: as .
Consider the sequence defined by:
where and .
The following proposition allows us to construct an ordinary least squares (OLS) estimator of the Hurst exponent of a fBm process based on sample expectiles.
Let and be the th order sample expectiles for the filtered series and respectively. Here two positive functions are considered, namely: for and . We have:
Now applying the logarithmic transformation to both sides of (9), we get:
On the other hand, (10) can be reformulated in the following way:
It is noteworthy here that we expect that and to converge towards 0 as . Hence, based on equations (11) and (12), we opt for an OLS regression scheme. This allows to derive the two following estimators of the hurst index defined by:
where is the vector of length with components , for some whereas for some vector of length designates the norm defined by . Notice here that and do not depend on .
We would like to put the stress on the fact that (13) and (14) are really similar to the ones developed 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 will be 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 or the trimmed-means of . 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.
Let a filter with order , , then as , and converge in probability to . Moreover, the following convergences in distribution hold
and where the matrices and are defined by (8).
Proof. We only provide a sketch of the proof. We claim that once Theorem 1 and Corollary 2 are established, the obtention of convergences stated in Proposition 4 are semi-routine. First of all, let us notice that
Since the functions and have Hermite rank 2 and since the correlation function of the stationary sequence decreases hyperbolically with an exponent then for any , Theorem 1 holds with (since for all ). This ensures the convergence in probability of the new estimates.
The cross-correlation between and is defined by
where according to (8), the matrices and are respectively defined by