Spectral estimation in the presence of missing data
Authors:
Natalia Bahamonde and Paul Doukhan
Journal:
Theor. Probability and Math. Statist. 95 (2017), 59-79
MSC (2010):
Primary 60; Secondary 60F, 62M10, 62M15
DOI:
https://doi.org/10.1090/tpms/1022
Published electronically:
February 28, 2018
MathSciNet review:
3631644
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Additional Information
Abstract: In this article we propose a quasi-Whittle estimator for parametric families of time series models in the presence of missing data. This estimator extends results to the incompletely observed case. This extension is valid to non-Gaussian and nonlinear models. It also allows us to bound the variance of an associated quasiperiodogram. A simulation study empirically validates the proposed estimate for mixing and nonmixing models.
References
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- Paul Doukhan, Nathanaël Mayo, and Lionel Truquet, Weak dependence, models and some applications, Metrika 69 (2009), no. 2-3, 199–225. MR 2481921, DOI https://doi.org/10.1007/s00184-008-0216-1
- Paul Doukhan, Gilles Teyssière, and Pablo Winant, A ${\rm LARCH}(\infty )$ vector valued process, Dependence in probability and statistics, Lect. Notes Stat., vol. 187, Springer, New York, 2006, pp. 245–258. MR 2283258, DOI https://doi.org/10.1007/0-387-36062-X_11
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- Murray Rosenblatt, Stationary sequences and random fields, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 885090
- Daniel Straumann, Estimation in conditionally heteroscedastic time series models, Lecture Notes in Statistics, vol. 181, Springer-Verlag, Berlin, 2005. MR 2142271
- P. Whittle, Gaussian estimation in stationary time series, Bull. Inst. Internat. Statist. 39 (1962), no. livraison 2, 105–129 (English, with French summary). MR 162345
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References
- D. Andrews, Non strong mixing autoregressive processes, J. Appl. Probab. 21 (1984), 930–934. MR 766830
- J. Bardet, P. Doukhan, and J. León, Uniform limit theorems for the periodogram of weakly dependent time series and their applications to Whittle’s estimate, J. of Time Series Anal. 29 (2008), no. 1, 906–945. MR 2450902
- P. Bondon and N. Bahamonde, Least squares estimation of arch models with missing observations, Journal of Time Series Analysis 33 (2012), no. 6, 880–891. MR 2991906
- R. Dahlhaus, A likelihood approximation for locally stationary processes, Annals of Statistics 28 (2000), 1762–1794. MR 1835040
- J. Dedecker and P. Doukhan, A new covariance inequality and applications, Stochastic Process. Appl. 106 (2003) no. 1, 63–80. MR 1983043
- P. Doukhan, Mixing: Properties and Examples, Lecture Notes in Statistics, vol. 85, Springer-Verlag, New York, 1994. MR 1312160
- P. Doukhan and G. Lang, Evaluation for moments of a ratio with application to regression estimation, Bernoulli 15 (2009), no. 4, 1259–1286. MR 2597592
- P. Doukhan and J. R. León, Cumulants for stationary mixing random sequences and applications to empirical spectral density, Probab. Math. Statist. 10 (1989), no. 1, 11–26. MR 990396
- P. Doukhan and S. Louhichi, A new weak dependence condition and applications to moment inequalities, Stochastic Process. Appl. 84 (1999), no. 2, 313–342. MR 1719345
- P. Doukhan, P. Massart, and E. Rio, The functional central limit theorem for strongly mixing processes, Ann. Inst. H. Poincaré Probab. Statist. 30 (1994), no. 1, 63–82. MR 1262892
- P. Doukhan, N. Mayo, and T. Truquet, Weak dependence, models and some applications, Metrika 69 (2009), 199–225. MR 2481921
- P. Doukhan, G. Teyssière, and P. Winant, A $\textrm {LARCH}(\infty )$ vector valued process, Dependence in Probability and Statistics, Lecture Notes in Statist., vol. 187, 2006, Springer, New York, pp. 245–258. MR 2283258
- W. Dunsmuir and P. M. Robinson, Asymptotic theory for time series containing missing and amplitude modulated observations, Sankhyā Ser. A 43 (1981), no. 3, 260–281. MR 665872
- W. Dunsmuir and P. M. Robinson, Estimation of time series models in the presence of missing data, J. Amer. Statist. Assoc. 76 (1981), no. 375, 560–568.
- S. Efromovich, Efficient non-parametric estimation of the spectral density in the presence of missing observations, Journal of Time Series Analysis 35 (2014), 407–427. MR 3259304
- R. F. Engle, Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation, Econometrica 50, no. 4, 987–1007. MR 666121
- L. Giraitis and P. M. Robinson, Whittle estimation of arch models, Econometric Theory 17 (1982), no. 3, 608–631. MR 1841822
- M. I. Knight, M. A. Nunes, and G. P. Nason, Spectral estimation for locally stationary time series with missing observations, Stat. Comput. 22 (2012), no. 4, 877–895. MR 2913790
- M. Lavielle, Detection of multiple changes in a sequence of dependent variables, Stochastic Processes and Appl. 83, no. 1, 79–102. MR 1705601
- E. Parzenm, On spectral analysis with missing observations and amplitude modulation, Sankhyā Ser. A 25 (1999), 383–392. MR 0172435
- E. Rio, Théorie asymptotique des processus aléatoires faiblement dépendants, Collection Mathématiques & Applications, vol. 31, Springer, 2000. MR 2117923
- M. Rosenblatt, A central limit theorem and a strong mixing condition, Proc. Natl. Acad. Sci. USA 42, 43–47. MR 0074711
- M. Rosenblatt, Stationary Sequences and Random Fields, Birkhäuser, New York, 1956. MR 885090
- D. Straumann, Estimation in Conditionally Heteroscedastic Time Series Models, Lecture Notes in Statistics, vol. 181, Springer-Verlag, Berlin, 2005. MR 2142271
- P. Whittle, Gaussian estimation in stationary time series, Bull. Inst. Internat. Statist. 39 (1962), no. 2, 105–129. MR 0162345
- Y. Yajima and H. Nishino, Estimation of the autocorrelation function of a stationary time series with missing observations, Sankhyā Ser. A 61 (1999), no. 2, 189–207. MR 1714870
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Additional Information
Natalia Bahamonde
Affiliation:
Instituto de Estadística, Pontificia Universidad Católica de Valparaíso
Address at time of publication:
Avenida Errazuriz 2734, Valparaíso, Chile
Email:
natalia.bahamonde@pucv.cl
Paul Doukhan
Affiliation:
UMR 8088 AGM, University Cergy-Pontoise, France
Address at time of publication:
Mathematics, office E 5.28, UCP site Saint-Martin, 2 Bd. Adolphe Chauvin, 95000 Cergy-Pontoise, France
Email:
doukhan@u-cergy.fr
Keywords:
Limit theorems,
time series,
auto correlation,
Whittle estimator,
weakly dependent
Received by editor(s):
September 6, 2016
Published electronically:
February 28, 2018
Additional Notes:
This work has been developed within the MME-DII center of excellence (ANR-11-LABEX-0023-01) and the Mathamsud 16-MATH-03 SIDRE project
Article copyright:
© Copyright 2018
American Mathematical Society
