Abstract

U-statistics has been widely studied and used in both statistics and machine learning. One challenge in application of U-statistics is the intensively demanding computation. In this thesis, we propose subsampling method to fast compute U-statistics. Our work is fourfold: (1) we formally accommodate uniform subsampling to fast computing of U-statistics; (2) we propose A-optimal subsampling method, which outperforms uniform subsampling method in terms of MSE; (3) we provide a method to approximate the A-optimal subsampling probabilities, since the running time of the A-optimal subsampling probabilities is the same as the full sample U-statistics; (4) we get the limiting distribution of the subsampling estimator. Then we run simulations and employ two real datasets to assess the performance of the uniform subsampling and the A-optimal subsampling methods. Our simulation and real data result shows that the MSE of A-optimal subsampling estimator is significantly smaller that of the uniform subsampling estimator. And the A-optimal subsampling estimator takes much less computing time than the full sample U-statistics if the subsample size is not too large compared to the full sample size.

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Committee Chair

Hanxiang Peng

Date of Award

8-2018

Committee Member 1

Benzion Boukai

Committee Member 2

Guang Lin

Committee Member 3

Zhongmin Shen

Committee Member 4

Fei Tan

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