# Robust Dickey-Fuller tests based on ranks for time series with additive outliers

V.A. Reisen, C. Lévy-Leduc, M. Bourguignon and H. Boistard,

to appear in Metrika.

In this paper the unit root tests proposed by Dickey and Fuller (DF) and their rank counterpart suggested by Breitung and Gouri ́eroux (1997) (BG) are analytically in- vestigated under the presence of additive outlier (AO) contaminations. The results show that the limiting distribution of the former test is outlier dependent, while the latter one is outlier free. The finite sample size properties of these tests are also investigated under different scenarios of testing contaminated unit root processes. In the empirical study, the alternative DF rank test suggested in Granger and Hallman (1991) (GH) is also considered. In Fotopoulos and Ahn (2003), these unit root rank tests were analytically and empirically investigated and compared to the DF test, but with outlier-free processes. Thus, the results provided in this paper complement the studies of the previous works, but in the context of time series with additive outliers. Equivalently to DF and Granger and Hallman (1991) unit root tests, the BG test shows to be sensitive to AO contaminations, but with less severity. In practical situations where there would be a suspicion of additive outlier, the general con- clusion is that the DF and Granger and Hallman (1991) unit root tests should be avoided, however, the BG approach can still be used.

# Large sample behavior of some well-known robust estimators under long-range dependence

C. Lévy-Leduc, H. Boistard, E. Moulines, M. S. Taqqu and V. A. Reisen.

Article published in Statistics, vol. 45, n. 1, p. 59–71, 2011.

The paper concerns robust location and scale estimators under long-range dependence, focusing on the Hodges-Lehmann location estimator, on the Shamos-Bickel scale estimator and on the Rousseeuw-Croux scale estimator. The large sample properties of these estimators are reviewed. The paper includes computer simulation in order to examine how well the estimators perform at finite sample sizes.

# Robust estimation of the scale and of the autocovariance function of Gaussian short and long-range dependent processes

C. Lévy-Leduc, H. Boistard, E. Moulines, M. S. Taqqu and V. A. Reisen.

Article published in Journal of Time Series Analysis, vol. 32, n. 2, p. 135-156, 2011. Download a pdf version.

A desirable property of an autocovariance estimator is to be robust to the presence of additive outliers. It is well-known that the sample autocovariance, being based on moments, does not have this property. Hence, the use of an autocovariance estimator which is robust to additive outliers can be very useful for time-series modeling. In this paper, the asymptotic properties of the robust scale and autocovariance estimators proposed by Rousseeuw and Croux (1993) and Ma and Genton (2000) are established for Gaussian processes, with either short-range or long-range dependence. It is shown in the short-range dependence setting that this robust estimator is asymptotically normal at the rate $\sqrt{n}$ , where n is the number of observations. An explicit expression of the asymptotic variance is also given and compared to the asymptotic variance of the classical autocovariance estimator. In the long-range dependence setting, the limiting distribution displays the same behavior than that of the classical autocovariance estimator, with a Gaussian limit and rate $\sqrt{n}$ when the Hurst parameter H is less than 3/4 and with a non-Gaussian limit (belonging to the second Wiener chaos) with rate depending on the Hurst parameter when $H\in (3/4, 1)$ . Some Monte- Carlo experiments are presented to illustrate our claims and the Nile River data is analyzed as an application. The theoretical results and the empirical evidence strongly suggest using the robust estimators as an alternative to estimate the dependence structure of Gaussian processes.

# Asymptotic properties of U-processes under long-range dependence

C. Lévy-Leduc, H. Boistard, E. Moulines, M. S. Taqqu and V. A. Reisen.

Article published in Annals of Statistics, vol. 39, n. 3, p. 1399-1426, 2011. Download a pdf version.

Let $(X_i)_{i\geq1}$ be a stationary mean-zero Gaussian process with covariances $\rho(k) = E(X_1X_{k+1})$ satisfying:
$\rho(0=1$ and $\rho(k)=k^{-D}L(k)$ where D is in (0,1) and L is slowly varying at infinity. Consider the U-process $\{U_n(r), r \in I\}$ defined as
$U_n(r)=\frac{1}{n(n-1)}\sum_{1\leq i\neq j \leq n}\mathbb{1}_{\{G(X_i,X_j\leq r\}},$
where I is an interval included in $\mathbb{R}$ and G is a symmetric function. In this paper, we provide central and non-central limit theorems for $U_n$ . They are used to derive, in the long-range dependence setting, new properties of many well-known estimators such as the Hodges- Lehmann estimator, which is a well-known robust location estimator, the Wilcoxon-signed rank statistic, the sample correlation integral and an associated robust scale estimator. These robust estimators are shown to have the same asymptotic distribution as the classical location and scale estimators. The limiting distributions are expressed through multiple Wiener-Itô integrals.