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Huber Estimation for Uncertain Autoregressive Model

School of Reliability and Systems Engineering, Beihang University, Beijing 100191, P. R. China

https://doi.org/10.1142/S1752890921500100Cited by:15 (Source: Crossref)

Abstract

Traditional time analysis deals with observations in chronological order assuming the observations are precise numbers under the framework of probability theory, whereas data are imprecisely collected in many cases. This paper characterizes the imprecisely observed data as uncertain variables and estimates the unknown parameters in the uncertain autoregressive model using Huber loss function, which is more flexible compared with other robust estimations for a pre-given $k$ that regulates the amount of robustness. Then prediction value and prediction interval of the future value are given. What is more, a method to choose $k$ by cross-validation is proposed. At last, numerical examples show our methods in detail and illustrate the robustness of Huber estimation by comparing it with the least square estimation. Our methods are also applied to a set of real data with carbon dioxide concentrations.

Keywords:

References

1. G. Walker , On periodicity in series of related terms, Proc. R. Soc. Lond. Ser. A 131 (1931) 518–532. Crossref, Google Scholar
2. G. Box and G. Jenkins , Time Series Analysis: Forecasting and Control (Holden-Day, San Francisco, 1970). Google Scholar
3. D. Cochrane and G. Orcutt , Applications of least squares regression to relationships containing autocorrelated errors, J. Am. Stat. Assoc. 44 (1949) 32–61. Crossref, Google Scholar
4. R. Shumway and W. Dean , Best linear unbiased estimation for multivariate stationary processes, Technometrics 10 (1968) 523–534. Crossref, Google Scholar
5. R. Jones , Maximum likelihood fitting of ARMA models to time series with missing observations, Technometrics 22 (1980) 389–395. Crossref, Google Scholar
6. P. J. Huber , Robust estimation of a location parameter, Ann. Math. Stat. 35 (1964) 73–101. Crossref, Google Scholar
7. D. Kahneman and A. Tversky , Prospect theory: An analysis of decision under risk, Econometrica 47 (1979) 263–292. Crossref, Google Scholar
8. B. Liu , Uncertainty Theory, 2nd edn. (Springer-Verlag, Berlin, 2007). Crossref, Google Scholar
9. B. Liu , Some research problems in uncertainty theory, J. Uncertain Syst. 3 (2009) 3–10. Google Scholar
10. K. Yao and B. Liu , Uncertain regression analysis: An approach for imprecise observations, Soft Comput. 22 (2018) 5579–5582. Crossref, Google Scholar
11. W. Lio and B. Liu , Residual and confidence interval for uncertain regression model with imprecise observations, J. Intell. Fuzzy Syst. 35 (2018) 2573–2583. Crossref, Google Scholar
12. Z. Liu and Y. Yang , Least absolute deviations estimation for uncertain regression with imprecise observations, Fuzzy Optim. Decis. Mak. 19 (2020) 33–52. Crossref, Google Scholar
13. Z. Liu and L. Jia , Uncertain Chapman-Richards growth model with imprecise observations, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 28 (2020) 769–783. Link, Google Scholar
14. X. Chen , Variation analysis of uncertain stationary independent increment process, Eur. J. Oper. Res. 222 (2012) 312–316. Crossref, Google Scholar
15. X. Chen , American option pricing formula for uncertain financial market, Int. J. Oper. Res. 8 (2011) 32–37. Google Scholar
16. X. Yang and B. Liu , Uncertain time series analysis with imprecise observations, Fuzzy Optim. Decis. Mak. 18 (2019) 263–278. Crossref, Google Scholar
17. X. Yang, G. Park and Y. Hu, Least absolute deviations estimation for uncertain autoregressive model, Technical Report (2019). Google Scholar
18. H. Tang , Uncertain vector autoregressive model with imprecise observations, Soft Comput. 24 (2020) 523–534. Crossref, Google Scholar
19. H. Tang, Uncertain threshold autoregressive model with imprecise observations, Commun. Stat. Theory Methods, doi:10.1080/03610926.2021.1906433. Google Scholar
20. D. Chen and X. Yang , Maximum likelihood estimation for uncertain autoregressive model with application to carbon dioxide emissions, J. Intell. Fuzzy Syst. 40 (2021) 1391–1399. Crossref, Google Scholar
21. D. Chen and X. Yang , Ridge estimation for uncertain autoregressive model with imprecise observations, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 29 (2021) 37–55. Link, Google Scholar
22. Z. Liu and X. Yang , Cross validation for uncertain autoregressive model, Commun. Stat. Simul. Comput. 23 (2020) 18211–18217. Google Scholar
23. T. Ye and X. Yang , Analysis and prediction of confirmed cases of COVID-19 in China by uncertain time series, Fuzzy Optim. Decis. Mak. 20 (2021) 209–228. Crossref, Google Scholar
24. B. Liu , Uncertainty Theory, 4th edn. (Springer-Verlag, Berlin, 2015). Crossref, Google Scholar
25. G. Yule , On a method of investigation periodicities in disturbed series with special reference to Wolfer’s sunspot numbers, Philos. Trans. R. Soc. Lond. 226 (1927) 267–298. Crossref, Google Scholar