Leverage Sampling for Single-Index Models

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In this thesis, a generalized leverage-based sub-sampling method for single-index models is proposed. The approach gives more efficient estimators than random sub-samples of the same size. Also, robust rank-based estimators of single-index models using leverage sub-samples provide estimators that are robust to outliers and heavy tails. A common bottleneck for rank-based estimators is the lack of computational efficiency, which is overcome using sub- samples. A simulation study was performed and, as expected the rank-based index direction estimator was comparable to the least squares index direction estimator when the errors follow a normal distribution. However, the rank-based index direction estimator was more efficient when the data followed a heavy-tailed error distribution. Finally, the results from a real data example are presented to highlight the performance of the proposed estimators.