  Robust M-estimation uses loss functions, such as least absolute deviation
(LAD), quantile loss and Huber's loss, to construct its objective function, in
order to for example eschew the impact of outliers, whereas the difficulty in
analysing the resultant estimators rests on the nonsmoothness of these losses.
Generalized functions have advantages over ordinary functions in several
aspects, especially generalized functions possess derivatives of any order.
Generalized functions incorporate local integrable functions, the so-called
regular generalized functions, while the so-called singular generalized
functions (e.g. Dirac delta function) can be obtained as the limits of a
sequence of sufficient smooth functions, so-called regular sequence in
generalized function context. This makes it possible to use these singular
generalized functions through approximation. Nevertheless, a significant
contribution of this paper is to establish the convergence rate of regular
sequence to nonsmooth loss that answers a call from the relevant literature.
For parameter estimation where objective function may be nonsmooth, this paper
first shows as a general paradigm that how generalized function approach can be
used to tackle the nonsmooth loss functions in Section two using a very simple
model. This approach is of general interest and applicability. We further use
the approach in robust M-estimation for additive single-index cointegrating
time series models; the asymptotic theory is established for the proposed
estimators. We evaluate the finite-sample performance of the proposed
estimation method and theory by both simulated data and an empirical analysis
of predictive regression of stock returns.
