Huber Estimation for Uncertain Autoregressive Model

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 [Formula: see text] 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 [Formula: see text] 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.