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

Fatimah A. Almulhim, Mohammed B. Alamari, Mustapha Rachdi () and Ali Laksaci
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Fatimah A. Almulhim: Department of Mathematical Sciences, College of Sciences, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
Mohammed B. Alamari: Department of Mathematics, College of Science, King Khalid University, Abha 62529, Saudi Arabia
Mustapha Rachdi: AGEIS Laboratory, Université Grenoble Alpes, UFR SHS, BP. 47, CEDEX 09, F-38040 Grenoble, France
Ali Laksaci: Department of Mathematics, College of Science, King Khalid University, Abha 62529, Saudi Arabia

Mathematics, 2024, vol. 12, issue 24, 1-17

Abstract: 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.

Keywords: functional data; ergodic data; recursive estimate; financial risk; expected shortfall; complete consistency; expectile regression; Kernel method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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