### LECTURE 3

### TWO TIMES WHEN THE COOKBOOK ISN'T ENOUGH

By now maybe you're feeling pretty good. The least-squares algorithms
which I outlined
in the preceding two lectures are very powerful and, to the extent
practical under the current
circumstances, I have tried to build them up from mathematical first
principles. Even where
I have waved my arms around and tried to make common-sense arguments,
mathematically
rigorous proofs can be found in any number of real statistics texts. It
seems as though these
least-squares methodologies represent the one true path to the best
answers in almost any
data-reduction problem you could imagine. That just isn't the
case. Least-squares may
be the part-time statistician's dream, because the algorithm is
straightforward and easily
applied, and it carries the endorsement of numerous texts on "Statistics
for Scientists."
Thus, the method of least squares apparently allows the astronomer to
claim before the
astronomical community and (more important?) before the referee, that
the very best that
can be done with the data *has* been done. Not so! These algorithms,
heavily documented
and widely used though they be, are not up to handing a couple of simple
problems
that we encounter every day when reducing real data. In fact, I believe
I can say with
some possibility of being right, that modern astronomers use the method
of least squares *improperly* more often than not.

There are two situations in particular that I would like to discuss
now. First, I have
mentioned once or twice that the development of these least-squares
solutions rests upon
the assumption that only *one* of the variables can contain error, and
that this variable can
be isolated on the left side of the equals sign. If this condition is
not met - and it very
often isn't - the equations we have used so far are invalid. Second,
real data do not in
general have a truly Gaussian error distribution. In fact, in almost
every case that you
will encounter in your careers, the *true* error distribution is not even
known. Again, blind
application of the algorithms we have developed so far will not
guarantee the "best" answer;
they may not even guarantee an answer that is "close enough for
government work"; I will
try to convince you that under some real-life circumstances they will
not even guarantee a *unique* answer.

Let's get to it.