As was described in the introduction to modelling in Section 1.10, a major task in statistics is the matching of a model to a set of data. The parameters of the model have to be estimated in such a way as to give a good fit between data and fitted model. The question that has to be asked is, “What do we mean here by good?” There are many answers to this question and hence there are many different approaches to the process of estimation. It is convenient to look at the methods to be discussed in this chapter as two general types. First, there are methods that seek to match specific population properties of the fitted model with the corresponding sample properties of the data. The low order population moments of the fitted model may, for example, be made equal to those of the sample. Second, there are methods based on minimizing some measure of the discrepancy between the fitted model and the data. In Chapter 1, this approach was illustrated with the minimization of the discrepancy measured by the sum of squares or absolute values of the distributional residuals. Most of the common methods of estimation have been developed for use with distributions defined by their density function or cumulative distribution function. We need to show how to fit a model defined by its quantile function. In the process of doing this, we will discover that some methods are particularly suited to distributions defined in terms of