Abstract
Several techniques for resampling dependent data have already been proposed. In this paper we use missing values techniques to modify the moving blocks jackknife and bootstrap. More specifically, we consider the blocks of deleted observations in the blockwise jackknife as missing data which are recovered by missing values estimates incorporating the observation dependence structure. Thus, we estimate the variance of a statistic as a weighted sample variance of the statistic evaluated in a “complete” series. Consistency of the variance and the distribution estimators of the sample mean are established. Also, we apply the missing values approach to the blockwise bootstrap by including some missing observations among two consecutive blocks and we demonstrate the consistency of the variance and the distribution estimators of the sample mean. Finally, we present the results of an extensive Monte Carlo study to evaluate the performance of these methods for finite sample sizes, showing that our proposal provides variance estimates for several time series statistics with smaller mean squared error than previous procedures.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abraham, B. and Thavaneswaran, A. (1991). A nonlinear time series model and estimation of missing observations,Ann. Inst. Statist. Math.,43, 493–504.
Berk, K. N. (1974). Consistent autoregressive spectral estimates,Ann. Statist.,2, 489–502.
Beveridge, S. (1992). Least squares estimation of missing values in time series,Comm. Statist. Theory Methods,21, 3479–3496.
Bose, A. (1990). Bootstrap in moving average models,Ann. Inst. Statist. Math.,42, 753–768.
Bosq, D. (1996).Nonparametric Statistics for Stochastic Processes: Estimation and Prediction, Springer, New York.
Bühlmann, P. (1995). Moving-average representation of autoregressive approximations,Stochastic Process. Appl.,60, 331–342.
Bühlmann, P. (1996). Locally adaptive lag-window spectral estimation,J. Time Ser. Anal.,17, 247–270.
Bühlmann, P. (1997). Sieve bootstrap for time series,Bernoulli,8, 123–148.
Bühlmann, P. and Künsch, H. R. (1994). Block length selection in the bootstrap for time series, R.R. No. 72, Eidgenössische Technische Hochschule.
Carlstein, E. (1986). The use of subseries values for estimating the variance of a general statistics from a stationary sequence,Ann. Statist.,14, 1171–1194.
Carlstein, E., Do, K., Hall, P., Hesterberg, T. and Künsch, H. R. (1998). Matched-block bootstrap for dependent data,Bernoulli,4, 305–328.
Efron, B. (1979). Bootstrap methods: Another look at the jackknife,Ann. Statist.,7, 1–26.
Efron, B. and Tibshirani, R. J. (1986). Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy,Statist. Sci.,1, 54–77.
Freedman, D. A. (1984). On bootstrapping two-stage least-square estimates in stationary linear models,Ann. Statist.,12, 827–842.
Galbraith, R. F. and Galbraith, J. I. (1974). On the inverse of some patterned matrices arising in the theory of stationary time series,J. Appl. Probab.,11, 63–71.
Hannan, E. J. and Kavalieris, L. (1986). Regression, autoregression models,J. Time Ser. Anal.,7, 27–49.
Harvey, A. C. and Pierse, R. G. (1984). Estimating missing observations in economic time series,J. Amer. Statist. Assoc.,79, 125–131.
Horn, R. A. and Johnson, C. R. (1990).Matrix Analysis, Cambridge University Press, Cambridge.
Kreiss, J. P. and Franke, J. (1992). Bootstrapping stationary autoregressive moving average models,J. Time Ser. Anal.,13, 297–317.
Künsch, H. R. (1989). The jackknife and the bootstrap for general stationary observations,Ann. Statist.,17, 1217–1241.
Lahiri, S. N. (2002). On the jackknife-after-bootstrap method for dependent data and its consistency properties,Econometric Theory,18, 79–98.
Liu, R. Y. and Singh, K. (1992). Moving blocks jackknife and bootstrap capture weak dependence,Exploring the Limits of Bootstrap (eds. R. Lepage and L. Billard), 225–248, Wiley, New York.
Peña, D. (1990). Influential observations in time series,J. Bus. Econom. Statist.,8, 235–241.
Peña, D. and Maravall, A. (1991). Interpolation, outliers, and inverse autocorrelations,Comm. Statist. Theory Methods,20, 3175–3186.
Politis, D. N. and Romano, J. F. (1992). A circular block-resampling procedure for stationary data,Exploring the Limits of Bootstrap (eds. R. Lepage and L. Billard), 263–270, Wiley, New York.
Politis, D. N. and Romano, J. F. (1994a). Large sample confidence regions based on subsamples under minimal assumptions,Ann. Statist.,22, 2031–2050.
Politis, D. N. and Romano, J. F. (1994b). The stationary bootstrap,J. Amer. Statist. Assoc.,89, 1303–1313.
Politis, D. N. and Romano, J. F. (1995). Bias-corrected nonparametric spectral estimation,J. Time Ser. Anal.,16, 67–103.
Quenouille, M. (1949). Approximation test of correlation in time series,J. Roy. Statist. Soc. Ser. B,11, 18–84.
Schwarz, G. (1978). Estimating the dimension of a model,Ann. Statist.,6, 461–464.
Shao, Q.-M. and Yu, H. (1993), Bootstrapping the sample means for stationary mixing sequences,Stochastic Process. Appl.,48, 175–190.
Sherman, M. (1998). Efficiency and robustness in subsampling for dependent data,J. Statist. Plann. Inference,75, 133–146.
Tukey, J. (1958). Bias and confidence in not quite large samples,Ann. Math. Statist.,29, p. 614.
White, H. (1984).Asymptotic Theory for Econometricians, Academic Press, New York.
Wu, C. F. J. (1990). On the asymptotic properties of the jackknife histogram,Ann. Statist.,18, 1438–1452.
Author information
Authors and Affiliations
About this article
Cite this article
Alonso, A.M., Peña, D. & Romo, J. Resampling time series using missing values techniques. Ann Inst Stat Math 55, 765–796 (2003). https://doi.org/10.1007/BF02523392
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02523392