Recursive Estimation of the Expectile-Based Shortfall in Functional Ergodic Time Series

This paper considers the Recursive Kernel Estimator (RKE) of the expectile-based conditional shortfall. The estimator is constructed under a functional structure based on the ergodicity assumption. More preciously, we assume that the input-variable is valued in a pseudo-metric space, output-variable is scalar and both are sampled from ergodic functional time series data. We establish the complete convergence rate of the RKE-estimator of the considered functional shortfall model using standard assumptions. We point out that the ergodicity assumption constitutes a relevant alternative structure to the mixing time series dependency. Thus, the results of this paper allows to cover a large class of functional time series for which the mixing assumption is failed to check. Moreover, the obtained results is established in a general way, allowing to particularize this convergence rate for many special situations including the kernel method, the independence case and the multivariate case. Finally, a simulation study is carried out to illustrate the finite sample performance of the RKE-estimator. In order to examine the feasibility of the recursive estimator in practice we consider a real data example based on financial time series data.