The effective use of spatial information, that is, the geographic locations
of population units, in a regression model-based approach to small area estimation is
an important practical issue. One approach for incorporating such spatial information
in a small area regression model is via GeographicallyWeighted Regression (GWR).
In GWR, the relationship between the outcome variable and the covariates is characterised
by local rather than global parameters, where local is defined spatially. In
this paper, we investigate GWR-based small area estimation under the M-quantile
modelling approach. In particular, we specify an M-quantile GWR model that is a local
model for the M-quantiles of the conditional distribution of the outcome variable
given the covariates. This model is then used to define a bias-robust predictor of the
small area characteristic of interest that also accounts for spatial association in the
data. An important spin-off from applying the M-quantile GWR small area model is
that it can potentially offer more efficient synthetic estimation for out of sample areas.
We demonstrate the usefulness of this framework through both model-based as well
as design-based simulations, with the latter based on a realistic survey data set. The
paper concludes with an illustrative application that focuses on estimation of average
levels of Acid Neutralising Capacity for lakes in the Northeast of the USA.