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.