Testing And Measuring The Conditional Mean(in) Dependence For Functional Data

With the advancement of data collection methods and storage technology,the collected data are becoming more and more diverse and most of them have characteristics,such as curves and images.They can be viewed as functional data.Functional data analysis is widely used in many fields,such as medicine,meteorology and engineering.In this paper,we foucs on testing and measuring conditional mean(in)dependence for functional data.Specifically,we consider whether one functional variable(predictor variable)contributes to the conditional mean of the other functional variable(response variable).This content has more applications,such as variable selection,feature screening,and model testing.In this paper,we propose a new metric,called angle martingale divergence,by combining the projection method and the special integration in Hilbert space.It is used to measure the contribution of one variable to the conditional mean of the other variable.The metric has the following properties:(1)it is nonnegative and equals to zero if and only if the conditional mean independence holds.In other words,predictor variable doesn’t contribute to the conditional mean of response variable;(2)it is invariable under the linear transformation of the predictor;(3)it doesn’t need the moment condition for the predictor variable.Based on this metric,there are two statistics for testing conditional mean independence in the paper.And,their asymptotic properties are also given.The critical value of the first test statistic is determined by wild bootstrap,and the other is obtained from the limiting normal distribution.Further,some criteria are proposed for measuring conditional mean dependence.And it is shown that angle martingale divergence has these properties.Data simulations show that the method proposed in this paper has better power than existing methods.It makes full use of the characteristics and information for functional data.Finally,we apply the method proposed in this paper to a real data.