| Modeling high dimensional data has always been a challenging problem in statistics. During the past two decades, several approaches to modeling high dimensional data using various dimensionality reduction techniques have been introduced. One of the most important methods in this area is due to Hastie and Tibshirani (1990), who introduced the additive model approach to modeling high dimensional data. Consider a response variable Y and a set of predictors (X1, X 2,…, Xp), the additive model proposed by Hastie and Tibshirani (1990) is , where the represents a zero mean random error term. The model essentially assumes that the p-dimensional regression function can be written as a sum of p one-dimensional smooth functions, which reduces the dimensionality of the problem. The estimation of component functions of the model is carried out by applying local linear smoothers recursively. In this dissertation, a new approach to estimation of the component functions of an additive model is introduced, which involves recursive computation of the local conditional median of the residuals. This work extends the univariate smoothing technique based on the local median proposed by Bhattacharya and Gangopadhyay (1990). The proposed estimator is not affected by extreme values (outliers) in the dataset, and hence the procedure provides robust estimates of the component functions. In addition, this approach performs better in modeling unexpected discontinuities in a regression curve, and also when the error distribution is heavy-tailed. Thus the proposed method provides a superior alternative to the solution developed by Hastie and Tibshirani (1990) for additive models. |
