  This paper discusses different classes of loss models in non-life insurance
settings. It then overviews the class Tukey transform loss models that have not
yet been widely considered in non-life insurance modelling, but offer
opportunities to produce flexible skewness and kurtosis features often required
in loss modelling. In addition, these loss models admit explicit quantile
specifications which make them directly relevant for quantile based risk
measure calculations. We detail various parameterizations and sub-families of
the Tukey transform based models, such as the g-and-h, g-and-k and g-and-j
models, including their properties of relevance to loss modelling.
  One of the challenges with such models is to perform robust estimation for
the loss model parameters that will be amenable to practitioners when fitting
such models. In this paper we develop a novel, efficient and robust estimation
procedure for estimation of model parameters in this family Tukey transform
models, based on L-moments. It is shown to be more robust and efficient than
current state of the art methods of estimation for such families of loss models
and is simple to implement for practical purposes.
