>>The introductory statistics courses are at the level
>>of alchemy or astrology, and provide no help in understanding how to
>>do things sensibly, if they do not make it much harder to understand
>>eventually what is difficult enough anyhow.
>Now, you are awakening my interest, and I will state a question:
>What books should be studied, in what sequence, to build a thorough
>understanding of statistics?
[Rest of quoted text at the end.]
I wish that I could give a good answer. At most, I can provide a fair one.
The first item has nothing directly to do with statistics; it is necessary
to have an understanding of probability. This does not mean games of chance,
but the properties of probability measures and expectation.
The next step is to look at a decision problem. This is not well done
anywhere, Raiffa's _Decision Analysis_ barely lays it out, and most others,
such as Berger, jump in the middle. The essence is that
One must simultaneously consider all consequences of the
proposed action in all states of nature.
The merits of this approach are rather easily seen by those who have not
been grounded in statistical methods. Those who "know" statistics have
trouble with this; what they have been taught contradicts it; the Bayesians
have less trouble.
Now if one assumes that actions are taken in a self-consistent manner,
which means less than many assume, then there must be a "utility function"
and a "prior measure" such that the appropriate action is to maximize the
integral. I know of no better proof of this than in my paper in _Statistics
and Decisions_ 1987. The only non-elementary mathematics is the use of the
Hahn-Banach Theorem to get the final conclusion.
There are more detailed expositions; Raiffa's book, cited above, was written
for graduate students in management with little mathematics. There are
several books by Lindley on the subject. Berger starts out assuming the
loss and prior, with little justification, but does not spend much time
on the foundations, and assumes the losses as given. But the use of a
well-defined non-monetary loss function only traces back to the axiomatic
approach of von Neumann and Morgenstern, whose axioms, including some
unstated ones, are stronger than mine. I would say that my main contribution
to this was the observation that the loss times prior comes from using the
same axiomatic approach throughout.
>I am personally interested in the answer. I studied 6 years with math and
>physics in university a lot of years ago, majoring in pure math with good
>ratings. Then I did choose to work with teaching for a lot of years, and
>started, by accident, to teach a small introductory stat course for
>engineering and business students. (10% of a years work for the students.)
>I do not find the precision of introductory texts impressive. Of course, the
>math should not be heavy in these books, but seeking in 10-15 books without
>finding the definitions of heavily used terminology *does* annoy me. This was
>not the situation in the math texts I once studied.
>Well, I ended up being interested in statistics and probability, so by now
>I am reading again when the kids are in bed, just for the pleasure of it, a
>bit more than I am teaching the students. Of course, I feel that I should know
>a lot more, as I do in math, and this is my goal. By now, I enjoy Wonnacott
>and Wonnacott: Regression, a second course in statistics, and Richard W.
>Hamming: The Art of Probability for Scientists and Engineers. These books is
>not heavily hit by the critique posed earlier in this message.
>But I will be glad to learn what books more knowlegeable people would
>recommend.
>So, is anyone willing to describe paths from beginner to the level at which
>one can do professional work? (I do understand that the road is only partially
>covered by books, and that real work is necessary. But for now, I am asking
>for the book part.)
--
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hrubin@stat.purdue.edu (Internet, bitnet)
{purdue,pur-ee}!snap.stat!hrubin(UUCP)