### 4. SAMPLING AND PARAMETER ESTIMATION. THE MAXIMUM LIKELIHOOD
METHOD

Sampling is the experimental method by which information can be
obtained about the parameters of an unknown distribution. As is well
known from the debate over opinion polls, it is important to have a
representative and unbiased sample. For the experimentalist, this
means *not* rejecting data because they do not ``*look right*''. The
rejection of data, in fact, is something to be avoided unless there
are overpowering reasons for doing so.

Given a data sample, one would then like to have a method for
determining the *best* value of the true parameters from the data. The
*best* value here is that which minimizes the variance between the
estimate and the true value. In statistics, this is known as
*estimation*. The estimation problem consists of two parts: (1)
determining the best estimate and (2) determining the uncertainty on
the estimate. There are a number of different principles which yield
formulae for combining data to obtain a best estimate. However, the
most widely accepted method and the one most applicable to our
purposes is the principle of *maximum likelihood*. We shall very briefly
demonstrate this principle in the following sections in order to give
a feeling for how the results are derived. The reader interested in
more detail or in some of the other methods should consult some of the
standard texts given in the bibliography. Before treating this topic,
however, we will first define a few terms.