Bernoulli

  • Bernoulli
  • Volume 22, Number 3 (2016), 1770-1807.

Quantile spectral processes: Asymptotic analysis and inference

Tobias Kley, Stanislav Volgushev, Holger Dette, and Marc Hallin

Full-text: Open access

Abstract

Quantile- and copula-related spectral concepts recently have been considered by various authors. Those spectra, in their most general form, provide a full characterization of the copulas associated with the pairs $(X_{t},X_{t-k})$ in a process $(X_{t})_{t\in\mathbb{Z}}$, and account for important dynamic features, such as changes in the conditional shape (skewness, kurtosis), time-irreversibility, or dependence in the extremes that their traditional counterparts cannot capture. Despite various proposals for estimation strategies, only quite incomplete asymptotic distributional results are available so far for the proposed estimators, which constitutes an important obstacle for their practical application. In this paper, we provide a detailed asymptotic analysis of a class of smoothed rank-based cross-periodograms associated with the copula spectral density kernels introduced in Dette et al. [Bernoulli 21 (2015) 781–831]. We show that, for a very general class of (possibly nonlinear) processes, properly scaled and centered smoothed versions of those cross-periodograms, indexed by couples of quantile levels, converge weakly, as stochastic processes, to Gaussian processes. A first application of those results is the construction of asymptotic confidence intervals for copula spectral density kernels. The same convergence results also provide asymptotic distributions (under serially dependent observations) for a new class of rank-based spectral methods involving the Fourier transforms of rank-based serial statistics such as the Spearman, Blomqvist or Gini autocovariance coefficients.

Article information

Source
Bernoulli, Volume 22, Number 3 (2016), 1770-1807.

Dates
Received: July 2014
Revised: February 2015
First available in Project Euclid: 16 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.bj/1458132999

Digital Object Identifier
doi:10.3150/15-BEJ711

Mathematical Reviews number (MathSciNet)
MR3474833

Zentralblatt MATH identifier
1369.62245

Keywords
Blomqvist copulas Gini spectra periodogram quantiles ranks Spearman spectral analysis time series

Citation

Kley, Tobias; Volgushev, Stanislav; Dette, Holger; Hallin, Marc. Quantile spectral processes: Asymptotic analysis and inference. Bernoulli 22 (2016), no. 3, 1770--1807. doi:10.3150/15-BEJ711. https://projecteuclid.org/euclid.bj/1458132999


Export citation

References

  • [1] Ahdesmäki, M., Lähdesmäki, H., Pearson, R., Huttunen, H. and Yli-Harja, O. (2005). Robust detection of periodic time series measured from biological systems. BMC Bioinformatics 6 117.
  • [2] Berghaus, B., Bücher, A. and Volgushev, S. (2014). Weak convergence of the empirical copula process with respect to weighted metrics. Available at arXiv:1411.5888.
  • [3] Blomqvist, N. (1950). On a measure of dependence between two random variables. Ann. Math. Stat. 21 593–600.
  • [4] Brillinger, D.R. (1975). Time Series: Data Analysis and Theory. New York: Holt, Rinehart and Winston.
  • [5] Brockwell, P.J. and Davis, R.A. (1987). Time Series: Theory and Methods. Springer Series in Statistics. New York: Springer.
  • [6] Carcea, M. and Serfling, R. (2014). A Gini autocovariance function for time series modeling. Preprint, UT Dallas.
  • [7] Cifarelli, D.M., Conti, P.L. and Regazzini, E. (1996). On the asymptotic distribution of a general measure of monotone dependence. Ann. Statist. 24 1386–1399.
  • [8] Dahlhaus, R. (1988). Empirical spectral processes and their applications to time series analysis. Stochastic Process. Appl. 30 69–83.
  • [9] Dahlhaus, R. and Polonik, W. (2009). Empirical spectral processes for locally stationary time series. Bernoulli 15 1–39.
  • [10] Davis, R.A. and Mikosch, T. (2009). The extremogram: A correlogram for extreme events. Bernoulli 15 977–1009.
  • [11] Davis, R.A., Mikosch, T. and Zhao, Y. (2013). Measures of serial extremal dependence and their estimation. Stochastic Process. Appl. 123 2575–2602.
  • [12] Dette, H., Hallin, M., Kley, T. and Volgushev, S. (2015). Of copulas, quantiles, ranks and spectra: An $L_{1}$-approach to spectral analysis. Bernoulli 21 781–831.
  • [13] Ferguson, T.S., Genest, C. and Hallin, M. (2000). Kendall’s tau for serial dependence. Canad. J. Statist. 28 587–604.
  • [14] Fermanian, J.-D., Radulović, D. and Wegkamp, M. (2004). Weak convergence of empirical copula processes. Bernoulli 10 847–860.
  • [15] Gasser, T., Müller, H.-G. and Mammitzsch, V. (1985). Kernels for nonparametric curve estimation. J. R. Stat. Soc. Ser. B. Stat. Methodol. 47 238–252.
  • [16] Genest, C., Carabarín-Aguirre, A. and Harvey, F. (2013). Copula parameter estimation using Blomqvist’s beta. J. SFdS 154 5–24.
  • [17] Genest, C. and Rémillard, B. (2004). Tests of independence and randomness based on the empirical copula process. TEST 13 335–370.
  • [18] Giraitis, L. and Koul, H.L. (2013). On asymptotic distributions of weighted sums of periodograms. Bernoulli 19 2389–2413.
  • [19] Hagemann, A. (2013). Robust spectral analysis. Available at arXiv:1111.1965v2.
  • [20] Hájek, J. (1968). Asymptotic normality of simple linear rank statistics under alternatives. Ann. Math. Stat. 39 325–346.
  • [21] Hallin, M. (2012). Ranks. In Encyclopedia of Environmetrics, 2nd ed. (W. Piegorsch and A. El Shaarawi, eds.) 2135–2152. New York: Wiley.
  • [22] Hallin, M. and Puri, M.L. (1992). Rank tests for time series analysis: A survey. In New Directions in Time Series Analysis, Part I (E. P. D. Brillinger and M. Rosenblatt, eds.) 111–153. New York: Springer.
  • [23] Hallin, M. and Puri, M.L. (1994). Aligned rank tests for linear models with autocorrelated error terms. J. Multivariate Anal. 50 175–237.
  • [24] Han, H., Linton, O.B., Oka, T. and Whang, Y.-J. (2014). The cross-quantilogram: Measuring quantile dependence and testing directional predictability between time series. Available at papers.ssrn.com/sol3/papers.cfm?abstract_id=2338468.
  • [25] Hill, J.B. and McCloskey, A. (2013). Heavy tail robust frequency domain estimation. Available at http://www.econ.brown.edu/fac/adam_mccloskey/Research_files/FDTTQML.pdf.
  • [26] Hong, Y. (1999). Hypothesis testing in time series via the empirical characteristic function: A generalized spectral density approach. J. Amer. Statist. Assoc. 94 1201–1220.
  • [27] Hong, Y. (2000). Generalized spectral tests for serial dependence. J. R. Stat. Soc. Ser. B. Stat. Methodol. 62 557–574.
  • [28] Katkovnik, V. (1998). Robust M-periodogram. IEEE Trans. Signal Process. 46 3104–3109.
  • [29] Kendall, M.G. (1938). A new measure of rank correlation. Biometrika 30 81–93.
  • [30] Kleiner, B., Martin, R.D. and Thomson, D.J. (1979). Robust estimation of power spectra. J. Roy. Statist. Soc. Ser. B 41 313–351.
  • [31] Kley, T. (2014). Quantile-based spectral analysis in an object-oriented framework and a reference implementation in R: The quantspec package. Available at arXiv:1408.6755.
  • [32] Kley, T. (2014). quantspec: Quantile-based spectral analysis functions. R package version 1.0-1.99.
  • [33] Kley, T., Volgushev, S., Dette, H. and Hallin, M. (2015). Supplement to “Quantile spectral processes: Asymptotic analysis and inference.” DOI:10.3150/15-BEJ711SUPP.
  • [34] Klüppelberg, C. and Mikosch, T. (1994). Some limit theory for the self-normalised periodogram of stable processes. Scand. J. Stat. 21 485–491.
  • [35] Koenker, R. and Bassett, G. Jr. (1978). Regression quantiles. Econometrica 46 33–50.
  • [36] Koenker, R. and Xiao, Z. (2006). Quantile autoregression. J. Amer. Statist. Assoc. 101 980–990.
  • [37] Lee, J. and Rao, S.S. (2012). The quantile spectral density and comparison based tests for nonlinear time series. Available at arXiv:1112.2759v2.
  • [38] Li, T.-H. (2008). Laplace periodogram for time series analysis. J. Amer. Statist. Assoc. 103 757–768.
  • [39] Li, T.-H. (2012). Quantile periodograms. J. Amer. Statist. Assoc. 107 765–776.
  • [40] Li, T.-H. (2013). Time Series with Mixed Spectra: Theory and Methods. Boca Raton: CRC Press.
  • [41] Linton, O. and Whang, Y.-J. (2007). The quantilogram: With an application to evaluating directional predictability. J. Econometrics 141 250–282.
  • [42] Liu, W. and Wu, W.B. (2010). Asymptotics of spectral density estimates. Econometric Theory 26 1218–1245.
  • [43] Maronna, R.A., Martin, R.D. and Yohai, V.J. (2006). Robust Statistics: Theory and Methods. Wiley Series in Probability and Statistics. Chichester: Wiley.
  • [44] Mikosch, T. (1998). Periodogram estimates from heavy-tailed data. In A Practical Guide to Heavy Tails (Santa Barbara, CA, 1995) (R. A. Adler, R. Feldman and M. S. Taqqu, eds.) 241–257. Boston, MA: Birkhäuser.
  • [45] Nelsen, R.B. (1998). Concordance and Gini’s measure of association. J. Nonparametr. Stat. 9 227–238.
  • [46] Priestley, M.B. (1981). Spectral Analysis and Time Series: Multivariate Series, Prediction and Control. New York: Academic Press.
  • [47] Schechtman, E. and Yitzhaki, S. (1987). A measure of association based on Gini’s mean difference. Comm. Statist. Theory Methods 16 207–231.
  • [48] Schmid, F., Schmidt, R., Blumentritt, T., Gaißer, S. and Ruppert, M. (2010). Copula-based measures of multivariate association. In Copula Theory and Its Applications (P. Jaworski, F. Durante, W.K. Härdle, T. Rychlik, P. Bickel, P. Diggle, S. Fienberg, U. Gather, I. Olkin and S. Zeger, eds.). Lect. Notes Stat. Proc. 198 209–236. Heidelberg: Springer.
  • [49] Shao, X. (2011). Testing for white noise under unknown dependence and its applications to diagnostic checking for time series models. Econometric Theory 27 312–343.
  • [50] Shao, X. and Wu, W.B. (2007). Asymptotic spectral theory for nonlinear time series. Ann. Statist. 35 1773–1801.
  • [51] Skaug, H.J. and Tjøstheim, D. (1993). A nonparametric test of serial independence based on the empirical distribution function. Biometrika 80 591–602.
  • [52] Skowronek, S., Volgushev, S., Kley, T., Dette, H. and Hallin, M. (2014). Quantile spectral analysis for locally stationary time series. Available at arXiv:1404.4605.
  • [53] Tjøstheim, D. (1996). Measures of dependence and tests of independence. Statistics 28 249–284.
  • [54] van der Vaart, A.W. and Wellner, J.A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer Series in Statistics. New York: Springer.
  • [55] Wald, A. and Wolfowitz, J. (1943). An exact test for randomness in the non-parametric case based on serial correlation. Ann. Math. Stat. 14 378–388.
  • [56] Wu, W.B. and Shao, X. (2004). Limit theorems for iterated random functions. J. Appl. Probab. 41 425–436.

Supplemental materials

  • Supplement to “Quantile spectral processes: Asymptotic analysis and inference”. We provide details for the proof of part (ii) of Theorem 3.6, and proofs for Propositions 3.1, 3.2, and 3.4. Further, we prove results from Section A.4, namely Lemmas A.1–A.7.