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'''Maurice FR&Eacute;CHET'''

b. 10 September 1878 - d. 4 June 1973


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'''Summary'''. Fréchet, one of the founders of modern analysis, also made 
various original contributions to the probability calculus. He played an
important role in the organization of a European scientific community
in the area of probability and statistics.

René Maurice Fréchet was born at Maligny in the département of the
Yonne, into a Protestant family, which moved to Paris soon after.
As a pupil at the Lycée Buffon,
M. Fréchet was taught by Jacques Hadamard (1865-1963), before the
latter left Paris for Bordeaux in 1894.  In 1900, he was admitted
to the &Eacute;cole Normale Supérieure, the most selective of Parisian
Institutions of higher learning.  In 1906, he defended one of the
most dazzling French theses in mathematics of his time.  In it,
Fr&eacute;chet defined the concept of an abstract metric space, and
within this framework demonstrated the validity of Weierstrass's
theorem.  Fr&eacute;chet's thesis marks the beginnings of analysis on
appropriately structured abstract spaces, one of the
principal new directions in analysis of this century 
(Arboleda, 1980; Taylor, 1982-1987). 
Throughout his life, Fréchet retained his taste 
for the most general theories and concepts, and some of those which he
introduced were real strokes of genius.  Sometimes reproached
for not having  proved truly "difficult theorems", he
is often credited only with introducing new concepts, generalizing and
resetting problems in a broader new framework within which he could
prove ``easy theorems" enunciated for his own interest.  It is, however, to
Fr&eacute;chet in particular that we owe the first theory
of integration with respect to an ``abstract measure", in 1915.
Kolmogorov was to make use of this to axiomatize probability theory,
citing ``his master Fr&eacute;chet" explicitly as his source.  Whatever one's
views, Fr&eacute;chet's scientific activity over a period of 60 years
was considerable, amounting to over 300 publications covering all
areas of pure and applied mathematics, and including about ten 
important books.

After a period as a schoolteacher at the Besan&ccedil;on (in 1907) and
Nantes (in 1908) Lyc&eacute;es, he was appointed lecturer 
in the Faculty of Sciences at Rennes. Then in 1910, he became Professor in
the Faculty of Sciences at Poitiers, replacing Henri Lebesgue, who
had been appointed to Paris. Both appointments were due to the
direct intervention of [[Borel, Émile|Borel]].  After the First World War, during
which he served as an Anglo-French interpreter, Fr&eacute;chet was
appointed Professor in the Faculty of Sciences at the newly
liberated  City of Strasbourg, showcase of French science.  
Actuarial studies and statistics had been taught at Strasbourg for a
long time, within the framework of the German university, associated with names
such as [[Lexis, Wilhelm|Lexis]] and [[Bortkiewicz, Ladislaus von|von Bortkiewicz]]. It was
here that Fr&eacute;chet, conscious of his national and international
role, and meticulous in the discharge of his duties,
began to teach applied mathematics, statistics, actuarial studies, and
nomography.  His first statistical papers date back to these
years.  It was entirely to be expected that Fr&eacute;chet should be
called to Paris in 1928, when the Institut Henri Poincar&eacute; (IHP)
was created, to develop the teaching of 
probability under the leadership of Borel, its director and the
holder of the Chair in the Calculus of Probabilities and
Mathematical Physics at the Sorbonne.  It should be noted that
Borel held the directorship of the IHP from 1928 until his death in
1956.   Apart from Borel,
Fr&eacute;chet was at the time the only French university academic of 
international renown who
was interested in recent developments in 
probability and statistics.  Paul L&eacute;vy who had just published a
remarkable book on the subject in 1925 (L&eacute;vy, 1925) was a
Professor at the &Eacute;cole Polytechnique and never held a university
position.

Fr&eacute;chet was first appointed  lecturer in 
probability at the Sorbonne's  Rockefeller Foundation, and then from
the end of 1928 as Professor (without a Chair), was promoted
to the tenured Chair of General Mathematics in 1933 and to
the Chair of Differential and Integral Calculus in 1935.  Finally, at the
start of 1941, he succeeded Borel in the Chair of the Calculus of
Probabilities and Mathematical Physics until his retirement in
1949. In 1928 (and at least until 1935) Fr&eacute;chet was also put in charge of
lectures at the &Eacute;cole Normale Sup&eacute;rieure. It was in this
capacity, with Borel's blessing, that he directed a sizeable number
of young mathematicians towards research in probability, in
particular Doeblin, Fortet, Lo&egrave;ve, Ville, and others.

As soon as he reached Paris in 1928 Fr&eacute;chet directed his research
towards the new ``theory of chain events", namely [[Markov, Andrei Andreevich|Markov]] chains. and 
published the first mathematical synthesis on this topic in
1938.  It was also Fr&eacute;chet who initiated the study of ``random elements"
taking values in the most general spaces.

In the statistical field, the usual evaluation of his works was not
always enthusiastic, to which following severe comment of Harald
Cram&eacute;r
<ref> ``Half a century with probability theory: some
personal recollections", "The Annals of Probability </ref>
, '''4'''(1976),
509-546 (see p. 528).'' attests:

"In early years Fr&eacute;chet had been an outstanding mathematician,
doing pathbreaking work in functional analysis.  He had taken up
probabilistic work at a fairly advanced age, and I am bound to say
that his work in this field did not seem very impressive to me."

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Nevertheless, Fr&eacute;chet was the author of several interesting
statistical papers, among them one on the one-dimensional Cram&eacute;r-Rao
inequality, some contributions in econometrics, and in spatial
statistics.

His statistical researches came from a critical reflection on
the theory of errors.  In a series of lectures, papers and
discussions, particularly with Paul L&eacute;vy, between the
two World Wars, Fr&eacute;chet mainly attacked the excessive 
hegemony of the Gaussian
distribution in all areas of application of the theory of
errors, and provided alternative solutions.  He demonstrated
the usefulness
of the Laplace probability density, $(exp - |x|), -\infty<x<\infty$, and 
in connection
with it, the use of the median instead of the mean.  It
was within this framework that he wrote his comments on the "l'homme-médian'', in contrast, presumably, to [[Quetelet, Adolphe|Quetelet]]'s
''l'homme-moyen''.  He continued his deviationism in 1926, constructing
a theory of errors based on an alternative composition of
elementary errors: instead of adding them, he proposed taking
their maximum.  Using an anologue of P. Lévy's argument in his
work of 1925, Fr&eacute;chet proved some fundamental results for the
statistics of extreme values, in particular concerning 
one of the three
asymptotic max-stable distributions.  He took up the topic of
extremes again in 1947, on the occasion of 
the jubilee of [[Mises, Richard von|Richard von Mises]], another researcher in this area.
Meanwhile, he gave his unflagging support to [[Gumbel, Emil Julius|Gumbel]]'s research on extremes,
which the latter had begun after his arrival in France in 1932/33.

Other foci of Fréchet's interest in statistics were closely tied
to Gumbel, as their rich correspondence indicates: the
concentration of incomes, the correlation coefficient, contingency
tables with fixed marginals, etc. (Hertz, 1997).

A recently rediscovered insight was what has  been named
Fréchet Optimality in the theory of probability inequalities (Seneta
and Chen, 1996)

Fréchet was a member of the International Statistical Institute
from 1931, Honorary Life Member from 1959, and Vice President from 1960.
It was he who chaired the first four international
conferences  devoted entirely to probability and
its applications in Geneva (1937), Lyon (1948), Paris (1949),
Amsterdam (1954).

For about 30 years, from the mid 1920's to the end of the 1950's,
Fr&eacute;chet acted as an intermediary for communicating information
in probability and statistics, as his varied correspondence with
scholars in many countries, preserved in the Archives of
the Academy of Sciences, attests.  In particular, this repository contains
some of the rare surviving letters of Paul Lévy before the Second 
World War.  An open individual, Fréchet worked after the 
Second World War for
the integration of scholars from the Soviet block into the
International Statistical Institute, for the peaceful union of peoples,
and the spread of Esperanto as a
universal language.  Some of his works, particularly
in Japan, were published in Esperanto.

Fréchet was elected to the Academy of Sciences of the Institut de
France in 1956, occupying the seat left vacant by &Eacute;mile Borel's
death. 


====References====
{|
|-
|valign="top"|{{Ref|1}}||valign="top"|   Arboleda, L.C.(1980). ''Contribution à l'&eacute;tude des premi&egrave;res recherches topologiques (d'apr&egrave;s la correspondance et les publications de Maurice Fr&eacute;chet)'', Thesis, Paris, EHESS, 1980. 
|-
|valign="top"|{{Ref|2}}||valign="top"|   Dugu&eacute;, D. (1974). Maurice Fréchet,  ''International Statistical Review'', '''42''', 113-114. 
|-
|valign="top"|{{Ref|3}}||valign="top"|   Fr&eacute;chet, M. (1938). ''Méthode des fonctions arbitraires. Théorie des &eacute;v&egrave;nements en cha&icirc;ne dans le cas d'un nombre fini d'états possibles'', Gauthier-Villars, Paris. Second Edition, with a new Supplement and Note by Paul L&eacute;vy, ibid., 1952. 
|-
|valign="top"|{{Ref|4}}||valign="top"|   Hertz, S&eacute;bastien (1997). ''Emil Julius Gumbel (1891-1966) et la statistique des extr&ecirc;mes'', Thesis, Universit&eacute; de Lyon-1. 
|-
|valign="top"|{{Ref|5}}||valign="top"|   L&eacute;vy, Paul (1925). ''Calcul des probabilités'', Gauthier-Villars, Paris. 
|-
|valign="top"|{{Ref|6}}||valign="top"|   Seneta, E. and Chen, T. (1996). Fréchet optimality of upper  bivariate Bonferroni-type bounds. ''Theory of Probability and Mathematical Statistics,'' '''52''', 147-152. 
|-
|valign="top"|{{Ref|7}}||valign="top"|   Taylor, A. E. (1982-1987). A study of Maurice Fréchet. I: His early work on point set theory and the theory of functionals, ''Archive for Hist. of Exact Sci.'', '''27''' (1982), 233-295. II: Mainly about his work on  general topology 1909-1928, ''ibid.'', '''34''', (1985), 279-380. III: Fr&eacute;chet as Analyst,  ''ibid.'', '''37''', (1987),  25-76.
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<references />

Reprinted with permission from
Christopher Charles Heyde and Eugene William Seneta (Editors),
Statisticians of the Centuries, Springer-Verlag Inc., New York, USA.

[[Category:Statprob]]
[[Category:Biographical]]