Missed Chances


Lecture
delivered on the occasion of the acceptance of the position of
special professor in the application of mathematics
at the University of Leiden
on Friday 20th February 1987


by

DR. R.D. GILL




1987
UNIVERSITY OF LEIDEN

Rector magnificus, Sir,

Gentlemen, governors of this University, Dean, Sir, of the Faculty of
Mathematics and Natural Sciences,

Ladies and Gentlemen Curators of this chair,

Ladies and Gentlemen Members of the Board of Directors of the Leiden
University-Fund,

Ladies and Gentlemen Professors, Lecturers and Members of the Scientific and of
the Technical and Administrative Staff,

Ladies and Gentlemen students,

and also all who by your presence show your interest,

Most appreciated listeners,

Most of you won't have stopped to think that the appointment of a mathematical
statistician to a chair of the applications of mathematics is somewhat
paradoxical.  One would be inclined to think of statistics preeminently as an
application of mathematics.
Even so one can consider this situation paradoxical, and I hope to explain this
to you in this coming hour.  First of all I must point out to the
non-mathematicians amongst you that, just as in mathematics in general there is
a distinction between pure and applied mathematics, so in statistics there is a
distinction between mathematical statistics and applied statistics.  In the
former one occupies oneself with problems of theory, while in the latter one
applies oneself to the specific aspects of certain applications and tries to
use statistics in concrete cases for aims outside the discipline.  I will even
put the following proposition to you:  though mathematical statistics is
inspired by a certain category of applications of mathematics, it is primarily
an offshoot of pure mathematics.  It not only has its own dynamics, but also
its own esthetic criteria of generality, abstraction and depth.  Of course the
research of mathematical statisticians can have practical consequences--most of
them will believe th
Today I'll examine the classical antithesis between pure and applied
mathematics,  emphasising  of course the  position of statistics and
statisticians.  I'd like to underline the existence of this antithesis, the
fact that there can be a paradox in putting mathematical statistics amongst the
applications of mathematics, by quoting the famous pure mathematician Paul
Halmos who said : `applied mathematics is bad mathematics'.  Such a statement
coming from that corner may not be surprising;  but what to think of the fact
that one of my colleagues in mathematical statistics quoted it recently in his
inaugural speech2 without any sign of condemnation?
Of course this is rather an exaggeration.  One could deduce from this that
there are two easily discernable kinds of statisticians, while many of us can
behave differently on different occasions.  Sometimes this results in a split
personality, occasionally in an enriched one, but always a certain tension
between theory and practice is present.  I hope to show you that the
application of statistics in the real world, and its interaction with
statistical theory, and even more broadly with mathematics as a whole, is a
much more subtle matter than one would think from the facile distinction
between theory and practice.  Mathematical and applied statistics are much
closer than is usually recognised.  One could even speak of a close
embracemaybe this is the reason for the special fascination of the discipline.
Lets now examine some applications of statistics in order to look at the role
played there by mathematics, clarifying at the same time the proposition I
outlined above.  To do this I'll first have to tell you a bit more about
statistics and its aims.

We are confronted daily in the newspapers and on television with pronouncements
in which rightly or wrongly statistics are used.  Statistics is used for
drawing conclusions from observations or data in situations in which chance or
accident played so big a role that an equivocal conclusion is not possible.  It
is only possible to draw inexact conclusions.  Statisticians are now able to
characterise and minimalise the level of uncertainty.  (Today I leave out the
important sub-fields `descriptive statistics' and `data-analysis' in which one
can or wants to do without pronouncements about chance.)  The fact that
according to one expert the Challenger disaster had a chance of one in every
thousand launches while according to another this chance was as big as one in
twenty might have something to do with the margins of uncertainty which are
undoubtedly present in these two statements.  On the other hand I certainly
don't want to suggest that in a situation of such complexity a solid
statistical solution is possi
The importance of a model--a framework for thought if you like--can also be
illustrated by the reports in the newspapers of the number of extra deaths to
be expected as a consequence of the accident in a nuclear reactor in
Tsjernobyl.  This number has no meaning whatsoever or, to put it kindly, it can
be interpreted it in many ways.  Everyone happens to die exactly once, disaster
or no disaster.  Does one mean really:  an additional number of deaths within a
certain (relatively short) period?  Or does this number apply to the total
number of people who die earlier than would have been the case had the accident
not happened?  And wouldn't it it be relevant in that case how much sooner that
would be;  i.e. wouldn't the `amount of lost man-years' be a more sensible
quantity to use?
I hope to make clear to you later that statistics can not only be used to draw
some conclusions from a collection of data;  it also has a much more important
use, because it enables us by employing mathematical models to find out whether
the available data really are relevant to answer the questions put to the
statistician.  As often as not this is not the case unless we add all kinds of
extra assumptions.  A statistical model makes these assumptions explicit and
thereby open to discussion, sometimes even disputable.
To illustrate the aforesaid I'll give you three examples.  The first two will
be quite brief.  The last one will not only be longer but also include some
advanced mathematics.  The first example takes us to seventeenth century
Leiden.  In that time in the Low Countries not only a political revolution was
being consolidated, but also a mathematical one.  It was here Descartes had
just developed analytic geometry.  He hereby showed algebra and geometry to be
one;  Euclid's geometry can be constructed as ordinary algebra of ordinary
numbers.  This quantification of space not only unified mathematics for the
first time in Western history, but it also offered new tools with which to
apply it.  Cartesian mathematics played an essential part in the development of
the European mechanistic world-view that culminated in the nineteenth century
with Laplace, and which we have inherited.  Descartes' new mathematics was only
a part of his new and controversial philosophy and physics.
In Leiden classical mathematics, not much changed since Euclid, was practised
in the classical way.  Prince Maurits altered this in 1600 by founding a chair
in applied--in particular military--mathematics.  A novelty concerning this
private chair was that lectures had to be in Dutch instead of the then current
Latin.  In 1615 Frans van Schooten was appointed to this chair.  When he died
his son, who had the same name, succeeded him;  this younger van Schooten now
was an ardent follower and popularizer of Descartes.  Around this time
Descartes himself stayed some years in Leiden.
The influence of the younger van Schooten shows itself clearly in the work of
three of his pupils:  Johan de Witt, Johannes Hudde, and Christiaan Huygens.
The contributions to probability-theory of Huygens are most well-known--his
book De Ratiociniis in Ludu Aleae published in 1657 was the first and also,
thanks to the wide distribution brought about by Van Schooten, during fifty
years the standard text on probability-theory.  Today however, I'd like to tell
you something about the work of De Witt.  He applied statistics for the first
time in politics.  This also made him the first person to take
probability-theory out of the domain of games of chance.  (De Witt's main
contribution to mathematics was something entirely different, namely a treatise
on conic sections `a la analytic geometry.)
In 1671 De Witt, as Grandpensionary, considered it his task to collect funds
quickly to defend the Republic in the impending attack from other major
countries (England, France, and the bishoprics of Munster and Cologne).  In
those days, when a small public service offered little opportunity to
economize, a common way of raising money was to sell annuities:  for a certain
price one could buy a fixed annual income or interest for a certain person;
this interest was paid as long as that person lived.  The buyer usually
nominated a younger member of his family.  The current purchase-price was
fourteen times the amount one got paid annually, independently of the age of
the nominee.  Inflation was four percent per annum.  If one knows the
probability distribution of the remaining lifetime of nominees of a given age
and bases oneself on a known level of inflation, one can nowadays very easily
calculate the fair purchase-price.  In those days this was very complicated;
even the language to think in these terms was
De Witt presented his report about the value of annuities to the States
General.  The report reads like a mathematical discourse.  It starts with
Huygens' definition of the expectation of a random variable--Huygens had been
the first to define this in mathematical terms.  (The definition is formulated
in terms of the price of a ticket in an equivalent fair lottery.)  The report
continues by explaining how to calculate an expectation given the probability
distribution of the random variable.  Because of the lack of empirical data De
Witt constructs the probability distribution of remaining lifetime by making
some plausible suppositions.  He divides a human lifespan into four periods.
Within each of these the chances of the person dying are uniformly
distributed;  the chances in the second period (from 53 to 63) are bigger by a
factor one-and-a-half than the chances before that (from 3 to 53), in the third
period (from 63 to 73) by a factor two, and in the fourth (from 73 to 80) by a
factor three.
After some arithmatic De Witt calculates the rather large purchase-price of
sixteen times the annual income for a nominee of the age of three.  He
substantiates this controversial proposal by giving many reasons why this
calculation would even be on the lower side.  In this he is helped by the
possibility of working through the effect of variations in the parameters he
chose.  It is a most powerful discourse that uses all means (in particular a
small statistical analysis from the registers of annuities of Holland and
West-Friesland) to support his proposal.
The proposal was accepted, although De Witt's carreer was nearly over;  a year
later after his leaving office he and his brother were lynched by the mob in
the Hague.  His discourse on annuities became renowned all over Europe--it was
known to Bernouilli and Leibniz for instance.
One may wonder if the members of the States General weren't dazzled by so much
mathematics;  all the more so because at close reading of the report some
strange incongruities come to light.  The factors one-and-a-half, two and three
are derived from an argument concerning conditional risks:  for a person aged
58 the death-risk or force of mortality in the coming year would be
one-and-a-half times as big as that for of a person aged 40.  However, the
calculations are concerned with unconditional chances:  i.e. the chance of a
three-year-old dying in his 58th year would be one-and-a-half times the chance
of him dying in his 40th year.  What's more, the factors in the calculations
have suddenly changed into two-thirds, a half and one third.  This means
they're the reciprocals of the original ones!  For a long time these `minor
flaws' went unnoticed.  In a later correspondence with Hudde, then town-mayor
of Amsterdam, who analysed new empirical data, De Witt showed he was well aware
of the distinction between ag
De Witt's calculations are set up efficiently.  One can repeat them easily
choosing different values for the parameters;  this possibility obviously
influenced his choice of model.  The original factors in the unconditional
probability distribution result in the politically unrealistic price of
eighteen annual interests.  It rather appears he chose his figures with the
wanted result in mind:  after first having got an unacceptable result he
adapted his figures without in his great haste (let's assume that not only
money but also time was scarce) following through the whole argument and
adapting it accordingly.  The lack of a clear and concise mathematical language
too hampered his study.  The matter was complicated even more by a mathematical
problem:  determining the risk-function over the ages up to a constant of
proportionality as De Witt wanted to do, does not fix the entire probability
distribution.  This does happen to be the case when determining the probability
density up to proportionality.
In the long run this work of De Witt hardly influenced the evolution of
statistics.  During the decades after his discourse was published the
mathematicians of Europe were busy developing the differential and integral
calculus.  De Witt's solution to the problem of annuities fell into oblivion.
It was only rediscovered two centuries later, during which many completely
incorrect solutions were used.  Probability-theory and statistics only
flourished in our century, when they could take root in much more mature
mathematics and be stimulated by prestigious scientific applications.
For the modern pursuit of science it's difficult to learn lessons from such a
history.  We can appreciate how De Witt applied with flourish the abstract
theory of games of chance, which had only just been developed, to a political
matter of great urgency.  The fact that his analysis was not perfect may be
defended in the light of an insufficient mathematical language and incomplete
conceptual apparatus.  Whether he himself fell victim to this opportunity for
confusion or whether he used it for political purposes, I happily leave to
historians to decide.
Let's continue with a more contemporary example:  the flooding of the
South-West of Holland in 1953.  On account of this disaster the Delta Committee
was formed.  The Mathematical Centre in Amsterdam was assigned the task of
determining by a statistical analysis of the levels of the high tides over the
past seventy years how high the seadikes should be--the `basispeil' (basic
level)--to make the chance of inundation one in ten thousand per annum,  or, to
put it differently and make the chance appear less negligible, one-hundredth in
every hundred years.  Since a statistical analysis in those days was literally
pure brain- and handwork, one had to be very sparing and resourceful in
constructing and analysing different models.  By selecting for the analysis one
high tide level for every depression occurring in the observational period, one
could rule out mutual dependence in the levels of high tides in close
succession.  Choosing these depressions however was a time- consuming and
somewhat subjective business.
In a stormy meeting in the Treveszaal at the Binnenhof in the Hague the
mathematicians--D. van Dantzig and J. Hemelrijk--managed to have the original
4.5 meters changed into 5 meters.  This greatly surprised many members of the
committee, who wondered what on earth these statisticians thought they were
doing.
At the present time this investigation is being repeated.  It is helped by a
now thirty years longer series of observations, much more refined methods of
analysis, and supported by modern computer facilities.  In particular--partly
inspired by such applications--an extensive and more elegant theory  has
evolved concerning extremes in a series of mutually dependent random
quantities, i.e. a stochastic process.  In an interim report the scientists use
a more complicated model with an extra parameter;  as a result of this the
estimated basispeil turns out to be much lower--4.2 meters only (the chances of
increases drop with the level already attained, thus making very extreme
waterlevels more unlikely).  By including this extra parameter however, the
possible estimation error has grown a lot (and a safe `basispeil' stays the
same).  This may even lead to the following final conclusion:  such a demanding
extrapolation can't be made from the available data.
My third and somewhat more detailed example concerns an area in modern
statistics in which I myself have been involved, namely so-called survival
analysis.  This term--a eufemism as you'll realise later--is the collective
name for statistical methods that can be used for the analysis of observed
lengths of time between the beginning of a medical treatment and the failing of
that treatment;  such data are collected for instance to compare a new
treatment with an old clinical treatment of certain kinds of cancer.  One
should think of a clinical trial in a hospital in which during say five years
maybe one to two hundred new patients suffering from a certain illness are
admitted and treated.  For every one of these patients chance is made to decide
whether they'll be treated according to the new or the old therapy.  As this
eliminates the effects of other factors, one is thus enabled to make as
unbiased a comparison as possible.  We are thinking of illnesses that can't be
cured now, but can be kept under control
In the fifties and sixties cancer research soared.  Under Kennedy for instance
the fight against cancer was taken up just as thoroughly as the race to be the
first to have a man on the moon.  Not only medical scientists occupied
themselves with this research, but in their wake many statisticians got
involved too.  This confrontation with a new kind of statistical problems led
to a new flourishing of statistical theory.
Comparing the distribution of a quantity in two populations, from each of which
a random sample has been taken, is a standard exercise for an applied
statistician.  Anyone who at any time has taken a course in elementary
statistics, be it for physicists, psychologists or whatever, will have been
confronted with the parametric `Student's t-test' and its non-parametric
counterpart the `two-sample test' of Wilcoxon.  The qualifications `parametric'
and `non-parametric' point to the fact that the first test rests on heavy
assumptions of normal distributions in the two populations to be compared, i.e.
distributions of a specified shape so that only two numerical parameters (mean
and variance) suffice to fix them completely.  The second procedure however is
correct under minimal assumptions.  In the kind of medical research I'm talking
about here, usually non-parametric methods are used.  One wants to draw as
convincing as possible a conclusion that the new treatment is better or worse
as the case may be.
Thus far this does not seem to offer the statistician a new challenge.  What I
havent mentioned yet is the complicating phenomenom called censoring.
Obviously one wants make a decision about the relative merit of each treatment
as soon as possible.  This implies that at the point in time at which one has
to analyse the data, quite a number of patients will still be in remission;
the better the new treatment the larger that number will be.  It's also
possible that patients who are in remission withdraw from the trial or die from
a totally independent cause.  The observation of the survival time of these
patients is censored.  At a certain (observed) moment in time a veil is drawn
over their further history.
Just leaving all these cases out of one's analysis is inefficient at best, at
worst completely misleading.  It's most important to include all data, censored
or not, recognising the difference.
Initially--I'm thinking of the fifties here--many ad hoc adaptations of the
classical statistical methods of analysis mentioned above were devised.  The
only positive thing about these methods was that they supplied something that
could be used.  A breakthrough was brought about only by essentially new
methods that fully recognised the dynamics of the situation.  I must stress
immediately that it was applied statisticians, N. Mantel and W. Haenszel in
particular, using elementary mathematics, but with a strong and healthy
intuition, who introduced these new methods.
The basic idea is this.  It's no use comparing the number of patients in
remission (surviving) in the two groups at a certain length of time after
admission, for these numbers not only result from the ending of the remission,
but also from censoring:  the termination of the trial and other causes.  Let's
suppose we look at a certain point in time within the two groups of survivors
at that moment.  If the censoring really is independent, each patient in these
groups has the same conditional chance of leaving remission in a subsequent
small time-interval, given the fact that he or she belongs to the group of
survivors, as would have been the case without censoring.  So it is possible to
make a fair comparison between the two treatments for every such time-interval,
at least as far as the conditional chance of leaving remission given the event
that the patient has been in remission till then.  Finally one has to combine
this whole series of comparisons, working all the time conditionally on what
has happened up
I have to go into some technical details now to show that the solution of
Mantel and Haenszel entails a few quite unusual elements.  Consider a
time-interval during which a certain patient leaves remission which is so small
that in this interval this is the only event occuring.  Classify all patients
who are present at the beginning of the interval in a two way table:  per row,
treatment group 1 or 2;  per column, does or does not leave remission in this
period.  Calculate the (non squared) contribution of one of the four cells to
the well known chi-square test-statistic of independence (observed minus
expected number), add these contributions over all the available points in
time, square, divide by the sum of the expected numbers, and compare with the
chi-square distribution with one degree of freedom.
To anyone acqainted with statistical theory this procedure make a mysterious
impression, however healthy the initial philosophy was.  In each of our two by
two tables one column has a total of one--one person leaving remission.  We
learn however the practical rule that the asymptotics--the `large sample
approximation'--of the chi-square test-statistic only works if at least five
individuals per cell occur under the hypothesis of independence of
treatment-group and the leaving of remission.  Another objection is the fact
that all these comparisons are interdependent;  the more patients leave
remission now, the fewer patients are left to compare at a later point in
time.
Even though people weren't entirely happy with the justification of these
methods, they put up with them, especially after from completely different, but
just as bizarre reasonings (Fisher score test based on the marginal density of
the rank numbers of the observations in an imaginary but intuitively comparable
experimental design) the same solution amazingly emerged.  To tackle all kinds
of variations on the basic problem described here, more and more refined
methods of analysis were devised via ever more daring heuristics and
intuition.  The climax was D. R. Cox's regression model, introduced in 1972,
and for which he only three years later (informally) pin-pointed the underlying
idea.  All this took place mainly in the medical-statistical and applied
statistical specialist literature.  Mathematical statisticians on the whole
either ignored all this or viewed it with suspicion, though some of them
studied the new statistical methods by using classical mathematical-statistical
techniques.  In a classic case
In this informal mathematics attention has shifted away from the traditional
counting over individuals to the counting over points in time.  The formal
mathematics--in this case probability theory--needed to study this will have to
make this transition too.  A similar shift has taken place for
medical-statistical practice:  it is easier to apply statistics if we're
prepared in the first place to investigate the time dependent conditional risk
of leaving remission, rather than concentrating on the unconditional chances of
leaving remission at different times.
Only in 1975 was the mathematical-statistical theory discovered that was needed
for a satisfactory, elegant and complete account of these methods.  This was
given in the Berkeley thesis of the Norwegian statistician Odd Olai Aalen.  It
took at least another ten years to fully exploit and develop and it gave me
great pleasure to participate in this exciting process.  Aalen arrived from
Oslo with practical experience in analysing survival times:  his master's
thesis was concerned with an investigation of the survival time of the the
intra-uterine device.  He was backed by a sound knowledge of the theory of
stochastic processes (`one bloody thing after another'
as the famous English statistician R.A. Fisher once explained to a journalist),
in particular Markov processes, as used in statistical applications in
demography and pervading modern probability theory.  In Berkeley he came into
contact with a group of mathematicians, the Frenchmen Bremaud and Jacod among
others, who were busy applying a new theory of stochastic integrals--a
stochastic infinitesimal calculus--to problems of controlling and filtering
counting processes.  It had become evident that the idea of the time-dependent,
conditional intensity of new events in a random process was closely connected
with the basic idea in this theory:  the Doob-Meyer decomposition of a nice
stochastic process into a systematic (predictable) part and a so-called
martingale.  Aalen recognised these same elements in survival analysis and
discovered also that all kinds of statistically interesting quantities could be
described in a simple way in terms of this theory, i.e. as stochastic integrals
of predictable processes wit
I'll attempt to clarify these terms somewhat.  The term martingale stems from
Monte Carlo:  it is a gambling system in which one supposes that if in roulette
the ball has fallen on red in less than half of the rounds, the chances that
this will happen in the next rounds will be bigger.  In probability theory
however a martingale is the abstraction of the cumulative gain (as function of
time) in a fair game of chance.  The average of this is zero, independent of
gambling system or rule for setting stakes.  In our example the connection is
that if the two treatments are equally good, and if censoring is independent of
survival, the result at each point in time of who, if anyone, leaves remission,
is pure chance just as in roulette.  The test-statistic of Mantel and Haenszel
is the final gain in this game when using a certain rule of setting stakes.  If
one treatment is better than the other, it isn't a fair game and (on average) a
cumulative gain (or loss) will occur.  From martingale theory one can see how
bi
All this amounted to the fact that, almost ready-made, exactly the right
mathematical theory became available that translated the intuition of the first
appliers into tough, precise mathematical theorems.  Where necessary it also
readjusted this intuition:  certain aspects of it had led them astray.  This
mathematical theory had just been formed by Paul-Andr'e Meyer and his school in
Strasbourg.  For a long time it was quite inaccessible, made inpenetrable by so
much French abstraction and details.  All the more so as it had been designed
as pure mathematics--l'art pour l'art--and rested on deep and abstract results
from the potential theory of Choquet.  Only gradually, and helped by the
applications, the essentials of this theory became extracted.
Through the years more theory has been added that time and again could
immediately be used in the applied field.  Here I am thinking in particular of
the martingale central limit theorems of the Chilean R. Rebolledo and of
members of the Russian school around A.N. Shiryayev.  With these one can give
conditions under which all kinds of statistical quantities have an
approximately normal distribution.  The use of these theorems belongs to the
most indispensable part of daily statistical practice (in particular the
application of the method of Mantel and Haenszel).
Looking back it is not difficult to find all kinds of forerunners and
indications of this theory in the applied literature.  An interesting and very
explicit example is offered by a paper10 in a famous English biostatistical
journal in which the authors left out a justification using martingal theory
because this according to the editors would have been too difficult for the
ordinary readers, who of course were mainly applied statisticians.  This was a
missed chance indeed.  At the same time this paper also gives an example of the
derailing of healthy practical intuition--I mention this to stress the fact
that beautiful but difficult mathematics is not just a game, but on the
contrary, absolutely vital to supply clarity, precision, and the firm base for
the next soaring of intuition.
The application of this theory of martingals, counting processes and stochastic
integrals has, via a number of brilliant successes, led to a uniform treatment
of a whole range of methods in survival analysis and to the clear demarcation
of their applicability.  It also led to a pruning of the uncontrolled growth of
partially useful, partially barren concepts and theories.  Now one can with the
greatest ease study new methods in an existing theoretical framework;  it is
also possible to  transplant ideas from the limited area of survival-analysis
to all kinds of different situations where the intensity of events in time is
studied.  I could give examples from demography, epidemiology, ethology,
psychology, and econometrics.  This baroque and inaccessible abstract theory
has now become an extremely strong and intuitively quite manageable calculus.
Id like to make some additional remarks concerning this last example.  Firstly,
all this activity has stimulated other areas within statistics very much
indeed.  Thus the applied field offered a number of examples of semiparametric
models:  statistical models in which infinite dimensional parameters occur.  I
won't explain what this means here, but the point is that exactly at the right
moment survival analysis offered a number of specific and interesting examples
and thereby gave a strong impetus to a just evolving synthesis of parametric
and non-parametric methods in the very heart of mathematical statistics:
phenomena arose that demanded admittance into a general theory;  an
experimental laboratory was available the findings of which any selfrespecting
theory ought to be able to predict and explain.
As I mentioned before all this also has important consequences for other areas
of applications.  In particular it's now becoming possible to investigate in a
sound way the interaction between observational frame and studied object (i.e.
life-histories) in demography and epidemiology.  I am thinking here of the fact
that in these disciplines one is more often than not compelled to work with
observational material that is retrospective in character:  collected after the
events, and depending on the random developments one actually studies included
in the sample or not, with the resulting consequences for bias.
These developments also had their effect on the statistical tradition in
certain countries, in particular France and the Soviet Union.  These countries
have very strong probabilistic schools, but are relatively weak as far as
applied statistics are concerned.  These new possibilities to apply their `own'
pure mathematics caused a renewed interest in applied statistics.  In both
countries mathematicians of stature became involved in it.
Of course it is appropriate to say something here about the consequences for
cancer-research, which after all was the subject of this example.  We must
recognise the fact that statistics plays such a vital role here precisely
because of the failure to make a real medical breakthrough.  It is indeed a
fact that statistics is at its best under circumstances like that.  For the
time being small improvements in `average' survival time have to be traced and
proven bit by bit with great difficulty.  This way differences in optimal
treatment for different patients have been established.  It is true that other
factors, `quality of life' for instance sometimes undo these small gains.
I'd like to make a second remark to give you a proper perspective.  Let's ask
ourselves:  what did the application of the martingale theory in this case
really consist of?  What was applied?  Apart from the modelling of certain
phenomena, and the motivation of certain methods of data analysis (both
certainly rather important), this beautiful mathematics mainly supplied
approximations the accuracy of which cannot (that is, not yet) be determined
from the theory:  the fact for instance that the test-statistic is
asymptotically normally distributed, and in a certain sense asymptotically
optimal.  I don't want to suggest that such results can't be applied, on the
contrary;  but we must realise that this application isn't properly completed
yet.  In all situations of some complexity we'll always have to content
ourselves with approximate solutions to the problems really posed.  The theory
has to be supplemented with empirical research and practical experience.
This example illustrates the fact that the mathematical theorems of
mathematical statistics, however beautiful and profound, are hardly ever
applied in the narrow sense.  (In my opinion the same applies to many branches
of mathematics if more than trivial applications are at stake.) Mathematical
statistics offers thinking-apparatus, tools with which to analyse complex
problems and to reduce them by approximations to subproblems that can be
understood well and thus solved.  The varying demands of reality compels the
statistician to find an ever changing compromise between analysability and
realism of the mathematical modelling.  The fact that there never is one right
solution gives the mathematician the liberty to find the most far-reaching, the
most enlightening solution.  The inferential statistics I talked about comes
into its own on the borderline between determinism and chaos, where patterns
can only just be discerned.  Statistics gets really interesting only if so much
chance is involved, that a definit
 Let me, esteemed listeners, finally quote Dirichlet, who in 1852 thought the
 direction of the then modern analysis to be:  `the supplanting of calculations
 by ideas'.  In my opinion this should be the task of mathematical statistics
 even now.  The distinction between pure and applied mathematics with which I
 started my discourse, can instinctively be characterised by beautiful ideas on
 the one hand, and possibly usefull but dull and endless calculations on the
 other.  I hope to have shown you that this distinction is misplaced.  The
 point is to find in applied mathematics the ideas that make the calculations
 selfevident and clear, and to shift the limits of ability and knowledge as far
 as possible.

Having arrived at the end of my discourse, in the first place I'd like to
express my thanks to Her Majesty the Queen for the decision to authorise the
Foundation Leiden University-Fund to found this chair.  To the general board of
the Leiden University-Fund I am very grateful for having appointed me to this
chair, and particularly to see in me a worthy successor of the very learned
Zoutendijk.  Furthermore I'd like to thank all who have devoted themselves to
my appointment.
I would also like to thank here the members of the Subfaculty of Mathematics
and Computer Science, in particular in the Department of Applied Mathematics,
for the enthousiasm with which they welcomed me in their midst and for the
pleasant and stimulating cooperation I have experienced there for one year
already.
Now I'd like to thank a number of people personally who greatly influenced my
mathematical development in this country.
Professor Hemelrijk, after I finished my studies in Cambridge, I became member
of your Department of Mathematical Statistics of the then Mathematical Centre,
now Centre for Mathematics and Computer Science (CWI) in Amsterdam. You put
first the application of statistics in an irreproachable way in and for society
at large.  I am still trying to follow your thoroughness and instinct in
finding  in each case the essential core of an applied problem.
Professor Oosterhof, dear Cobus, as adviser of the department it was your fate
to supervise me in the research for my thesis, in a discipline neither of us at
the start knew anything about.  Your carefulness and commitment have been
immensely important to me.  You taught me how one really should supervise a
graduate student.
Professor Groeneboom, dear Piet, you were only half a year my senior at the MC
but I learned a lot from you.  Your perseverance to penetrate to the root of
any problem that seized you, however far it led you into unknown regions of
mathematics, still goes on inspiring me.
Professor van Zwet, dear Willem, your acumen, erudition, and swiftness of mind
are amazing for me and possibly for the rest of your environment.  It is a
privilege and a pleasure to be allowed to work with you here in Leiden.
Honoured ladies and gentlemen curators, directors and staff members of the CWI,
dear collleagues of the Department of Mathematical Statistics, I am most glad
that I am allowed to combine this chair with my work at the CWI.  To really be
a Centre, intensive contact with the university life of this country is vitally
important;  moreover to an institution that isn't in the first place
educational, the good training of researchers elsewhere is a matter of vital
importance.  The institute thanks its first-ranking position mainly to the
fact that the it works at problems with a long term view.  In statistics, that
is based on so broad a foundation, depth can only be reached through long and
concentrated study, and after a truly academic education.  I do hope that in
our somewhat pragmatic times, in which the eye mostly does not look further
than tomorrow, this long term view will meet response from our government.
Ladies and gentlemen students, you have heard that mathematical statistics is a
difficult subject that is of the greatest importance to society, even if it is
at times only to sober up.  In this discipline one has to develop a refined
taste and a great knowledge to choose the right path in the jungle of
possibilities for abstraction and thus analysis of a certain applied problem.
To apply statistics, but also to contribute to its theory, one has to be a
schooled mathematician with empathy for the objectives of each applied field.
I hope you will take your chance and accept this challenge with dedication.

I have spoken.


References

1. P.R. Halmos (1981), Applied mathematics is bad mathematics, Mathematics
tomorrow  (L.A. Steen, ed.), 9-20, Springer, New York.

2. W. Albers (1985), De ongrijpbare zekerheid (Unattainable certainty),
Inaugural lecture, University of Limburg, Maastricht.

3. J. de Witt (1671), Waerdye van Lyf-Renten naer Proportie van Los-Renten, The
Hague;  see also:  A. Hald (in preparation),  A history of probability,
statistics and insurance mathematics before 1880.

4. Cooperative research group `basispeilen' (1986), Onderzoek Basispeilen.
Interim report, June 1986,  report GWIO-86.008, Ministry of Public Works, Tidal
waters section, The Hague;  see also:  L. de Haan (1986), The arch-enemy
attacked with mathematics, Mathematics and Computer Science  II (M.
Hazewinckel, J.K. Lenstra & L.G.L.T. Meertens, eds), CWI Monograph 4, 51-60,
North-Holland.

5. N. Mantel & W. Haenszel (1959), Statistical aspects of the analysis of data
from retrospective studies of disease, J. Nat. Cancer Inst. 22, 719-748;  see
also:  N. Mantel (1966), Evaluation of survival data and two new rank order
statistics arising in its consideration, Cancer Chemotherapy Reports 50,
163-170.

6. R. Peto & J. Peto (1972), Asymptotically efficient rank invariant test
procedures (with discussion), J. Roy. Statist. Soc. (A) 135, 185-206.

7. D. R. Cox (1972), Regression models and life tables (with discussion), J.
Roy. Statist. Soc. (B) 34, 187-200.

8. J. Crowley & D.R. Thomas (1975), Large sample theory for the log rank test,
Tech. Report 415, Dept. of Statistics, Univ. of Wisconsin-Madison.

9. O. O. Aalen (1975), Statistical theory for a family of counting processes ,
Ph.D. thesis, University of California, Berkeley;  reprint (1976), Institute of
Mathematical Statistics, University of Copenhagen.

10. R.E Tarone & J. Ware (1977), On distribution free tests for the equality of
survival distributions, Biometrika 64, 156-160.

11. G.P. Lejeune Dirichlet (1852), Gedaechtenisrede auf C.G.J. Jacobi [p. 19],
Abh. Preuss. Akad. Wiss. 1-26;  see also:  E.E. Kummer (1860), Gedaechtenisrede
auf G.P. Lejeune Dirichlet [p. 33], Abh. Koenig. Akad. Wiss. Berlin 1-36;
and:  N. Bourbaki (1950), The architecture of mathematics [p.231], Amer. Math.
Monthly 57, 221-231.

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