Functional sufficient dimension reduction based on weighted method

Abstract

In the last 20 years, many of functional sufficient dimension reduction (FSDR) methods are developed. Among all of the FSDR methods, the most popular methods are functional sliced inverse regression (FSIR) and functional sliced average variance estimation (FSAVE). FSIR and FSAVE are based on the first and second moment of the functional predictor, which are not robust to outlier. A robust version of FSIR called the soft weighted FSIR (SFSIR) is proposed method, which used a trimmed and spatial median estimate to replace the classic moment estimation. The trimmed estimate is very useful and SFSIR method can deal with the outlier data very well. However, discarding the potential outlier usually leads to low efficient estimate. And the trimmed estimation method need to select two tuning parameters and to detect the potential outlier is very computational costing. Furthermore, FSIR and SFSIR methods are well-known for their poor performance when the link function is even (or symmetrically dependency). Then, a robust FSDR method with high efficient and low computation costing estimate is urgent needed in practical. In the presented paper, firstly, we will propose weighted based FSIR and FASVE methods. Secondly, we will also study the robust FSIR and FSAVE methods by series expansion, which first approximates the functional predictor by functional principal component basis, and then conduct FSDR methods on the coefficients. Numerical studies including simulation study and real data analysis show that our proposed weighted methods have wonderful performance.