Composite Quasi-likelihood For Single-index Models With Massive Datasets

The single-index models(SIMs)not only maintains the characteristics of non-parametric smoothness but also achieves dimensionality reduction of the data,that is,it projects the information of the p-dimensional covariate X to the one-dimensional variable XTγ.It retains the original explanatory nature and avoid the "curse of dimensionality".Many existing estimation procedures for SIMs were built on least square loss,which is popular for its mathematical beauty but is non-robust to non-normal errors and outliers.This paper addressed the question of both robustness and efficiency of estimation methods based on a new data-driven weighted linear combination of convex loss functions instead of only quadratic loss for SIMs.In Chapter 2,we study the related problems of composite quasi-likelihood,given the asymptotic properties of the estimator and its proof process,the calculation formula of the bandwidth and the selection method of weights to provide maximum efficiency,and these optimal weights can be estimated based on the data.The optimal weights can be chosen to provide maximum efficiency and these optimal weights can be estimated from data.In Chapter 3,as a specific example,we introduce a robust method of composite least square and least absolute deviation methods and apply it to a single index model.We give the specific steps for calculating the estimators of γLS-LAD and gLS-LAD(·)and the calculation process of the optimal weight.After obtaining the optimal weight,the asymptotic relative efficiency of our method is compared with that of MAVE and LAD,and the LS-LAD has the highest efficiency.In Chapter 4,we extend the proposed method to the analysis of massive datasets via a divide-and-conquer strategy.The proposed approach significantly reduces the required primary memory by dividing the data set into M subsets,and each subset is calculated at the same time and the resulting estimate is as efficient as if the entire dataset was analyzed simultaneously.In Chapter 5,the simulation studies and real data applications are conducted to illustrate the finite sample performance of the proposed methods.The actual data results are compared with the estimation results of the LS method and the LAD method to illustrate the advantages of the proposed method of fast calculation speed and high estimation accuracy.