![](https://csdnimg.cn/release/download_crawler_static/1823415/bg10.jpg)
xv PREFACE xv
of detrending or seasonal adjustment in Econometrics. But old data sets, if preserved unmutilated
by old assumptions, mayhave a new lease on life when our prior information advances.
Style of Presentation:
In part A, exp ounding principles and elementary applications, most
Chapters start with several pages of verbal discussion of the nature of the problem. Here we
try to explain the constructiveways of lo oking at it, and the logical pitfalls responsible for past
errors. Only then do we turn to the mathematics, solving a few of the problems of the genre to the
point where the reader may carry it on by straightforward mathematical generalization. In part B,
expounding more advanced applications, we can concentrate from the start on the mathematics.
The writer has learned from much experience that this primary emphasis on the logic of the
problem, rather than the mathematics, is necessary in the early stages. For modern students, the
mathematics is the easy part; once a problem has b een reduced to a denite mathematical exercise,
most students can solve it eortlessly and extend it endlessly, without further help from any book or
teacher. It is in the conceptual matters (how to make the initial connection b etween the real{world
problem and the abstract mathematics) that they are perplexed and unsure how to pro ceed.
Recent history demonstrates that anyone foolhardy enough to describ e his own work as \rig-
orous" is headed for a fall. Therefore, we shall claim only that we do not knowingly give erroneous
arguments. We are conscious also of writing for a large and varied audience, for most of whom
clarity of meaning is more imp ortant than \rigor" in the narrow mathematical sense.
There are two more, even stronger reasons for placing our primary emphasis on logic and
clarity. Firstly, no argument is stronger than the premises that go into it, and as Harold Jereys
noted, those who lay the greatest stress on mathematical rigor are just the ones who, lacking a sure
sense of the real world, tie their arguments to unrealistic premises and thus destroy their relevance.
Jereys likened this to trying to strengthen a building by anchoring steel b eams into plaster. An
argument which makes it clear intuitively
why
a result is correct, is actually more trustworthy
and more likely of a p ermanent place in science, than is one that makes a great overt showof
mathematical rigor unaccompanied by understanding.
Secondly,wehave to recognize that there are no really trustworthy standards of rigor in a
mathematics that has embraced the theory of innite sets. Morris Kline (1980, p. 351) came close
to the Jereys simile: \Should one design a bridge using theory involving innite sets or the axiom
of choice? Might not the bridge collapse?" The only real rigor wehavetoday is in the operations
of elementary arithmetic on nite sets of nite integers, and our own bridge will be safest from
collapse if wekeep this in mind.
Of course, it is essential that we follow this \nite sets" p olicy whenever it matters for our
results; but we do not propose to become fanatical about it. In particular, the arts of computation
and approximation are on a dierent level than that of basic principle; and so once a result is
derived from strict application of the rules, we allow ourselves to use any convenient analytical
methods for evaluation or approximation (such as replacing a sum byanintegral) without feeling
obliged to showhow to generate an uncountable set as the limit of a nite one.
But we impose on ourselves a far stricter adherence to the mathematical rules of probability
theory than was ever exhibited in the \ortho dox" statistical literature, in which authors repeatedly
invoke the aforementioned intuitive
ad hoc
devices to do, arbitrarily and imp erfectly, what the
rules of probability theory as logic would have done for them uniquely and optimally. It is just this
strict adherence that enables us to avoid the articial paradoxes and contradictions of ortho dox
statistics, as described in Chapters 15 and 17.
Equally important, this p olicy often simplies the computations in twoways: (A) The problem
of determining the sampling distribution of a \statistic" is eliminated; the evidence of the data is
displayed fully in the likelihood function, which can b e written down immediately. (B) One can
eliminate nuisance parameters at the b eginning of a calculation, thus reducing the dimensionality
of a search algorithm. This can mean orders of magnitude reduction in computation over what