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In practice, as opposed to a large set of finite-dimensional vectors approximated from discrete data, we often prefer to utilize functional data. In recent years, partially observable function data have frequently appeared in practical applications and are the objectofan increasing interest by the literature. In this thesis, we learn the concept of data integration depth of partially observable functions proposed by Elias et al. 2023 [4] , which can be used to measure the degree of data centralization. At the same time, we also studied the trimmed-mean estimator method and consistency proof proposed by Fraiman and Muniz 2001 [6] for completely observable functions. This method refers to the process of removing some of the smallest and largest values before calculating the mean to enhance the robustness of the estimate. In this thesis, we introduce the concept of trimmed-mean estimator for partially observable functions. We discuss several theoretical and practical issues, including the proof that the proposed trimmed-mean estimator converges almost surely and provide a simulation study. The results show that our estimator performs better in terms of efficiency and robustness compared to the ordinary mean under partially observable functional data.

Model checking methods based on non-parametric estimation are widely used because of their tractable limiting null distributions and being sensitive to high-frequency oscillation alternative models. However, this kind of test suffers from the curse of dimensionality, resulting in slow convergence, especially for functional data with infinite dimensional features. In this paper, we propose an adaptive-to-model test for a parametric functional single-index model by using the orthogonality of residual and its conditional expectation. The test achieves model adaptation by sufficient dimension reduction which utilizes functional sliced inverse regression. This test procedure can be easily extended to other non-parametric test methods. Under certain conditions, we prove the asymptotic properties of the test statistic under the null hypothesis, fixed alternative hypothesis and local alternative hypothesis. Simulations show that our test has better performance than the method that does not use functional sufficient dimension reduction. An analysis of COVID-19 data verifies our conclusion.