Robust estimation in semiparametric models

Assume that we have data coming from a semiparametric model. We consider robust estimation when we suspect that the data may have been contaminated. Following Hampel's approach, we focus on asymptotically linear estimates with bounded influence functions. The main object of our work is to find optimal influence functions which solve Hampel's variational problem and then use the optimal influence functions to construct the optimal B-robust estimates.;We have conducted simulation studies to the symmetric location models and the paired exponential mixture models. The empirical results that reveal that the optimal estimates behave reasonably well even in small sample situations.;For a general semiparametric model, we are able to identify the lowest bound that an influence function can have. We have established existence and uniqueness of the optimal influence function that solves Hampel's problem corresponding to a bound larger than the lowest one. As a special case of semiparametric models, a general multidimensional parametric model with nuisance parameters is investigated. We then establish the existence and uniqueness of explicit optimal influence functions as stated by Hampel et al. (1986). The general theory is applied to the semiparametric heteroscedastic regression models and the semiparametric mixture models. In these cases, the optimal influence functions have explicit expressions and the optimal estimates have been constructed.