Multivariate local polynomial fitting is applied to the multivariate linear heteroscedastic regression model. Firstly, the local polynomial fitting is applied to estimate heteroscedastic function, then the coefficients of regression model are obtained by using generalized least squares method. One noteworthy feature of our approach is that we avoid the testing for heteroscedasticity by improving the traditional two-stage method. Due to non-parametric technique of local polynomial estimation, it is unnecessary to know the form of heteroscedastic function. Therefore, we can improve the estimation precision, when the heteroscedastic function is unknown. Furthermore, we verify that the regression coefficients is asymptotic normal based on numerical simulations and normal Q-Q plots of residuals. Finally, the simulation results and the local polynomial estimation of real data indicate that our approach is surely effective in finite-sample situations.

Abstract Low-rank matrix approximation has applications in many fields, such as 3D reconstruction from an image se-quence and 2D filter design. In this paper, one issue with low-rank matrix approximation is re-investigated: the miss-ing data problem. Much effort was devoted to this prob-lem, and the Wiberg algorithm or the damped Newton algo-rithm were recommended in previous studies. However, the Wiberg or damped Newton algorithms do not suit for large (especially “long”) matrices, because one needs to solve a large linear system in every iteration. In this paper, we revi-talize the usage of the Levenberg-Marquardt algorithm for solving the missing data problem, by utilizing the prop-erty that low-rank approximation is a minimization prob-lem on subspaces. In two proposed implementations of the Levenberg-Marquardt algorithm, one only needs to solve a much smaller linear system in every iteration, especially for “long ” matrices. Simulations and experiments on real data show the superiority of the proposed algorithms. Though the proposed algorithms achieve a high success rate in estimat-ing the optimal solution by random initialization, as illus-trated by real examples; it still remains an open issue how to properly do the initialization in a severe situation (that is, a large amount of data is missing and with high-level noise).