The Annals of Statistics

Simultaneous nonparametric inference of time series

Weidong Liu and Wei Biao Wu

Full-text: Open access


We consider kernel estimation of marginal densities and regression functions of stationary processes. It is shown that for a wide class of time series, with proper centering and scaling, the maximum deviations of kernel density and regression estimates are asymptotically Gumbel. Our results substantially generalize earlier ones which were obtained under independence or beta mixing assumptions. The asymptotic results can be applied to assess patterns of marginal densities or regression functions via the construction of simultaneous confidence bands for which one can perform goodness-of-fit tests. As an application, we construct simultaneous confidence bands for drift and volatility functions in a dynamic short-term rate model for the U.S. Treasury yield curve rates data.

Article information

Ann. Statist., Volume 38, Number 4 (2010), 2388-2421.

First available in Project Euclid: 11 July 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H15: Hypothesis testing
Secondary: 62G10: Hypothesis testing

Gumbel distribution kernel density estimation linear process maximum deviation nonlinear time series nonparametric regression simultaneous confidence band stationary process treasury bill data


Liu, Weidong; Wu, Wei Biao. Simultaneous nonparametric inference of time series. Ann. Statist. 38 (2010), no. 4, 2388--2421. doi:10.1214/09-AOS789.

Export citation


  • Aït-Sahalia, Y. (1996a). Nonparametric pricing of interest rate derivative securities. Econometrica 64 527–560.
  • Aït-Sahalia, Y. (1996b). Testing continuous-time models of the spot interest rate. Rev. Finan. Stud. 9 385–426.
  • Berman, S. (1962). A law of large numbers for the maximum of a stationary Gaussian sequence. Ann. Math. Statist. 33 93–97.
  • Bickel, P. J. and Rosenblatt, M. (1973). On some global measures of the deviations of density function estimates. Ann. Statist. 1 1071–1095.
  • Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy 81 637–654.
  • Bosq, D. (1996). Nonparametric Statistics for Stochastic Processes. Estimation and Prediction. Lecture Notes in Statistics 110. Springer, New York.
  • Brillinger, D. R. (1969). An asymptotic representation of the sample distribution function. Bull. Amer. Math. Soc. 75 545–547.
  • Bühlmann, P. (1998). Sieve bootstrap for smoothing in nonstationary time series. Ann. Statist. 26 48–83.
  • Chan, K. C., Karolyi, A. G., Longstaff, F. A. and Sanders, A. B. (1992). An empirical comparison of alternative models of the short-term interest rate. J. Finance 47 1209–1227.
  • Chapman, D. A. and Pearson, N. D. (2000). Is the short rate drift actually nonlinear? J. Finance 55 355–388.
  • Courtadon, G. (1982). The pricing of options on default-free bonds. J. Finan. Quant. Anal. 17 75–100.
  • Cox, J. C., Ingersoll, J. E. and Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica 53 385–403.
  • Cummins, D. J., Filloon, T. G. and Nychka, D. (2001). Confidence intervals for nonparametric curve estimates: Toward more uniform pointwise coverage. J. Amer. Statist. Assoc. 96 233–246.
  • Doukhan, P. and Louhichi, S. (1999). A new weak dependence condition and applications to moment inequalities. Stochastic Process. Appl. 84 313–342.
  • Doukhan, P., Madre, H. and Rosenbaum, M. (2007). Weak dependence for infinite ARCH-type bilinear models. Statistics 41 31–45.
  • Doukhan, P. and Portal, F. (1987). Principe d’invariance faible pour la fonction de répartition empirique dans un cadre multidimensionnel et mélangeant. Probab. Math. Statist. 8 117–132.
  • Dümbgen, L. (2003). Optimal confidence bands for shape-restricted curves. Bernoulli 9 423–449.
  • Erdös, P. (1939). On a family of symmetric Bernoulli convolutions. Amer. J. Math. 61 974–976.
  • Fan, J. and Yao, Q. (1998). Efficient estimation of conditional variance functions in stochastic regression. Biometrika 85 645–660.
  • Fan, J. and Yao, Q. (2003). Nonlinear Time Series. Nonparametric and Parametric Methods. Springer, New York.
  • Fan, J. and Zhang, C. (2003). A re-examination of diffusion estimators with applications to financial model validation. J. Amer. Statist. Assoc. 98 118–134.
  • Freedman, D. A. (1975). On tail probabilities for martingales. Ann. Probab. 3 100–118.
  • Granger, C. W. J. and Joyeux, R. (1980). An introduction to long-memory time series models and fractional differencing. J. Time Ser. Anal. 1 15–29.
  • Györfi, L., Härdle, W., Sarda, P. and Vieu, P. (1989). Nonparametric Curve Estimation From Time Series. Springer, Berlin.
  • Härdle, W. and Marron, J. S. (1991). Bootstrap simultaneous error bars for nonparametric regression. Ann. Statist. 19 778–796.
  • Hall, P. and Titterington, D. M. (1988). On confidence bands in nonparametric density estimation and regression. J. Multivariate Anal. 27 228–254.
  • Ho, H. C. and Hsing, T. (1996). On the asymptotic expansion of the empirical process of long-memory moving averages. Ann. Statist. 24 992–1024.
  • Hosking, J. R. M. (1981). Fractional differencing. Biometrika 68 165–176.
  • Johnston, G. J. (1982). Probabilities of maximal deviations for nonparametric regression function estimates. J. Multivariate Anal. 12 402–414.
  • Knafl, G., Sacks, J. and Ylvisaker, D. (1985). Confidence bands for regression functions. J. Amer. Statist. Assoc. 80 683–691.
  • Komlós, J., Major, P. and Tusnády, G. (1975). An approximation of partial sums of independent RV’s and the sample DF. I. Z. Wahrsch. Verw. Gebiete 32 111–131.
  • Komlós, J., Major, P. and Tusnády, G. (1976). An approximation of partial sums of independent RV’s and the sample DF. II. Z. Wahrsch. Verw. Gebiete 34 33–58.
  • Neumann, M. H. (1998). Strong approximation of density estimators from weakly dependent observations by density estimators from independent observations. Ann. Statist. 26 2014–2048.
  • Robinson, P. M. (1983). Nonparametric estimators for time series. J. Time Ser. Anal. 4 185–207.
  • Rosenblatt, M. (1976). On the maximal deviation of k-dimensional density estimates. Ann. Probab. 4 1009–1015.
  • Stanton, R. (1997). A nonparametric model of term structure dynamics and the market price of interest rate risk. J. Finance 52 1973–2002.
  • Shao, X. and Wu, W. B. (2007). Asymptotic spectral theory for nonlinear time series. Ann. Statist. 35 1773–1801.
  • Sun, J. and Loader, C. R. (1994). Simultaneous confidence bands for linear regression and smoothing. Ann. Statist. 22 1328–1345.
  • Tjøstheim, D. (1994). Nonlinear time series: A selective review. Scand. J. Statist. 21 97–130.
  • Vasicek, O. A. (1977). An equilibrium characterization of the term structure. J. Financial Economics 5 177–188.
  • Wu, W. B. (2003). Empirical processes of long-memory sequences. Bernoulli 9 809–831.
  • Wu, W. B. (2005). Nonlinear system theory: Another look at dependence. Proc. Natl. Acad. Sci. USA 102 14150–14154.
  • Wu, W. B. and Mielniczuk, J. (2002). Kernel density estimation for linear processes. Ann. Statist. 30 1441–1459.
  • Xia, Y. (1998). Bias-corrected confidence bands in nonparametric regression. J. R. Stat. Soc. Ser. B Stat. Methodol. 60 797–811.
  • Zaĭtsev, A. Y. (1987). On the Gaussian approximation of convolutions under multidimensional analogues of S. N. Bernstein’s inequality conditions. Probab. Theory Related Fields 74 535–566.
  • Zhao, Z. (2008). Parametric and nonparametric models and methods in financial econometrics. Stat. Surv. 2 1–42.
  • Zhao, Z. and Wu, W. B. (2008). Confidence bands in nonparametric time series regression. Ann. Statist. 36 1854–1878.