A type of optimum online learning from quantile estimating equation and objective function

{The significance and novelty of this work are established through methodological development, theoretical analysis, and numerical experiments for quantile regression in streaming data.} Estimating equation and objective function are perhaps the most common tools in classical statistics and machine learning. The optimum frameworks are frequently used for optimizing the statistical inference and learning. Up to now, however, the optimum forms of the online updating have not been well explored. In this paper, these classical optimum notions are utilized to solve the modern issue: online learning. For quantile and quantile regression, a type of optimum frameworks of the online updating estimating equation and objective function are proposed, the notion of expectation derivative and the smoothing techniques are introduced to deal with non-differentiable objective functions, and the relevant iterative algorithms are suggested to implement the estimation procedures. The proposed methods can be applied to the models with possible heterogenous distributions. Theoretically, the asymptotic normality and oracle property are established. The behaviours of the methods are further illustrated by various numerical examples from simulation experiments and real data analysis.