Statistical inference for time series with non-precise data

Abstract For linear time series models, the asymptotic properties of least squares estimation are known. An analogous result for the stationary time series with non-precise data is considered in this paper. The least squares estimation for the general autoregressive model with fuzzy data is investigated with a suitable fuzzy metric. The fuzzy-type least squares approach is defined, and analogues of the conventional normal equations and fuzzy least squares estimators (FLSEs) are also derived. A numerical example for computing FLSEs and forecasting the future values of fuzzy series is given with the financial data. Asymptotic normality and consistency are established for the FLSEs. A confidence region based on a class of FLSEs is constructed. The asymptotic relative efficiency of FLSEs with respect to the crisp least squares estimators is provided. Some simulation results are also presented to illustrate the small sample behavior of FLSEs.

[1]  Gwo-Hshiung Tzeng,et al.  Fuzzy ARIMA model for forecasting the foreign exchange market , 2001, Fuzzy Sets Syst..

[2]  A. Wald,et al.  On the Statistical Treatment of Linear Stochastic Difference Equations , 1943 .

[3]  Jin Hee Yoon,et al.  A unified approach to asymptotic behaviors for the autoregressive model with fuzzy data , 2014, Inf. Sci..

[4]  L. H. Koopmans.  The spectral analysis of time series , 1974 .

[5]  Gwilym M. Jenkins,et al.  Time series analysis, forecasting and control , 1972 .

[6]  Frederico G. Guimarães,et al.  Short-term load forecasting method based on fuzzy time series, seasonality and long memory process , 2017, Int. J. Approx. Reason..

[7]  Rudolf Kruse.  The strong law of large numbers for fuzzy random variables , 1982, Inf. Sci..

[8]  Hsiao-Fan Wang,et al.  Fuzzy relation analysis in fuzzy time series model , 2005 .

[9]  Richard A. Davis,et al.  Time Series: Theory and Methods , 2013 .

[10]  Woo Joo Lee,et al.  A hybrid dynamic and fuzzy time series model for mid-term power load forecasting , 2015 .

[11]  harald Cramer,et al.  Stationary And Related Stochastic Processes , 1967 .

[12]  Phil Diamond,et al.  Fuzzy least squares , 1988, Inf. Sci..

[13]  Yuanjie Zheng,et al.  Long-term prediction of time series based on stepwise linear division algorithm and time-variant zonary fuzzy information granules , 2019, Int. J. Approx. Reason..

[14]  Zhi-Wen Zhao,et al.  Asymptotic properties of least squares estimation for a new fuzzy autoregressive model , 2013 .

[15]  P. Diamond,et al.  Extended fuzzy linear models and least squares estimates , 1997 .

[16]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[17]  M. Priestley.  Evolutionary Spectra and Non‐Stationary Processes , 1965 .

[18]  R. Serfling.  Approximation Theorems of Mathematical Statistics , 1980 .

[19]  Mahua Bose,et al.  Designing fuzzy time series forecasting models: A survey , 2019, Int. J. Approx. Reason..

[20]  Shyi-Ming Chen,et al.  Forecasting enrollments based on fuzzy time series , 1996, Fuzzy Sets Syst..

[21]  Gwo-Hshiung Tzeng,et al.  A fuzzy seasonal ARIMA model for forecasting , 2002, Fuzzy Sets Syst..

[22]  Abraham Kandel,et al.  General fuzzy least squares , 1997, Fuzzy Sets Syst..

[23]  Harald Cramér,et al.  On Harmonic Analysis in Certain Functional Spaces , 1992 .

[24]  Ying Li,et al.  Asymptotic properties of least squares estimation with fuzzy observations , 2008, Inf. Sci..

[25]  B. Chissom,et al.  Fuzzy time series and its models , 1993 .

[26]  Liangjian Hu,et al.  Analysis of dynamical systems whose inputs are fuzzy stochastic processes , 2002, Fuzzy Sets Syst..

[27]  B. Chissom,et al.  Forecasting enrollments with fuzzy time series—part II , 1993 .

[28]  M. Puri,et al.  Fuzzy Random Variables , 1986 .

[29]  Witold Pedrycz,et al.  Long-term forecasting of time series based on linear fuzzy information granules and fuzzy inference system , 2017, Int. J. Approx. Reason..