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\noindent{\bf Thorvald Nicolai THIELE}\\
b. 24 December 1838 -  d. 26 September 1910
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\noindent{\bf Summary.} A man of many talents, Professor (at the Copenhagen
Observatory) T.N. Thiele was a leading personality in astronomy, numerical
analysis, actuarial science, and mathematical statistics. He invented the
cumulants and discovered Thiele's differential equation.
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Thorvald Nicolai Thiele was born in Copenhagen on December 24,
1838. His father was an intellectual and cultural leader who
belonged to a prominent Danish family.
Thorvald Nicolai (named after his
godfather, the famous Danish sculptor and benefactor of arts,
B. Thorvaldsen) was a gifted child. His interest in mathematics was
stimulated
by Professor C. J{\o}rgensen, who directed him to university studies,
not in pure mathematics, but in astronomy, which J{\o}rgensen considered
more apt to accommodate his protege's diverse talents.
Thiele obtained his master's degree in astronomy in 1860 and his doctoral
degree in 1866, both from the University of Copenhagen. His
thesis for the doctorate, on orbits in double star systems, was written during
a period as assistant to Professor H.L.
d'Arrest at the Copenhagen Observatory, where he served from 1860-1870. 

During 1871-1872 Thiele worked out the actuarial basis for the life
insurance company Hafnia, founded in 1872 with Thiele as Mathematical
Director (actuary) until 1901. This was the typical arrangement in those 
days; the
actuarial profession was still in its infancy, and the analytical skills
of astronomers, physicists, and other applied mathematicians made them
wizards in insurance and various other fields of quantitative study.
In 1885 Thiele was elected corresponding member of the British Institute of
Actuaries (founded 1848) and in 1895 he became a member of the Board of
the newly established Permanent Committee of the International Congresses
of Actuaries. He took the initiative to found the Danish
Actuarial Society in 1901 and became its first President, performing
this task until his death.
In 1875 Thiele was appointed Professor of Astronomy and Director of the
Copenhagen Observatory. He became a member of The Royal Danish Society
(Videnskaberne Selskab) in 1879, served as Rector of the University of
Copenhagen 1900-1906, and retired in 1907. He died on September 26, 1910,
in Copenhagen.  Thiele had married Marie Martine Trolle (1841-1889) in
1867 and they had six children.

By the interplay of local circumstances and traditions of the era, Thiele
developed exceptional theoretical skills, not only in his speciality, but
also in numerical analysis, mathematical statistics, and actuarial science,
invariably combined with deep involvement in practical applications. In
particular, he stressed the importance of empirical model testing by
analysis of residuals, and he was an expert in numerical calculation.
A (presumably complete) list of Thiele's written works with 52 entries is
provided by Nielsen (1912). He authored three monographs; two on statistical
inference (Thiele, 1889, 1897) - the second a simplified version of the
first - and one on interpolation theory (Thiele, 1909).

Key references on Thiele's contributions to
mathematical statistics and actuarial science, respectively, are the
authoritative works of Hald (1981, 1998) and Hoem (1983).
A short biography, with independent views, is found
in Schweder (1980). As with Johnson and Kotz (1997), we 
present only a brief account.

Thiele's interest in statistics arose naturally from his empirical
studies in astronomy and mortality investigations, where errors of 
observations play a significant role. He
contributed to the theory of skew distributions.
He formulated the canonical form of the linear model with normally
distributed errors and reduced the general linear model to canonical form
by orthogonal
transformations, and he made early fundamental contributions to the
analysis of variance and dynamical linear models (time series).
He invented the cumulants and investigated their properties in a series
of papers. Hald (1998) suggests that his
motivation may have been on the practical and computational side: Moments
of high orders are numerically intractable since they essentially grow
exponentially with their order; the cumulants, defined as the coefficients in
the series expansion of the logarithm of the moment generating function,
are stable. The Gram-Charlier series might just as well 
have been called the Thiele-Charlier series.
Due to imperfect communication,
and also a sad tradition of mutual neglect across borders, Thiele's 
contributions to statistics were not widely
recognized and cited by his British contemporaries. The translation
of his 1897 monograph into English was poor
and has served only partly to earn Thiele appropriate acknowledgement,
although it was later reprinted in full in {\it Annals of Mathematical
Statistics} in 1931.

In actuarial science Thiele introduced
a mortality law capable of fitting mortality at all ages and, 
strongly influenced by L.H.F. Opperman, he was at the
forefront in the development of methods for graduation of mortality tables
(Hoem, 1980, 1983).
He also discovered the celebrated Thiele's differential equation which
is described below.

The premium reserve of a life insurance policy is defined, at any time during
the term of the policy, as the expected discounted value of future benefits
less premiums. For an $n$-year death insurance of $S$ against level premium
$P$ per time unit, issued at time 0 to an $x$ years old insured, the
reserve at time $t$ after issue is the solution to 
$$
\frac{d}{dt} V_t = P + \delta V_t - \mu_{x+t} (S - V_t) \,,
$$
subject to $V_n = 0$.
Here $\mu_y$ is the force of mortality at age $y$, $\delta$ is the
force of interest.
The tremendous impact of Thiele's differential equation on the development
of modern life insurance mathematics is emphasized by Berger (1939) and
Hansen (1946) and is apparent in e.g. the works of Hoem (1988) and Norberg
(1991).
It is the first rudiment of stochastic analysis
in continuous time life insurance mathematics, and as such plays a role
similar to Filip Lundberg's (q.v.) pioneering work in actuarial risk theory
some 30 years later.

It is a paradox that, among Thiele's many
fundamental contributions to the statistical sciences, the differential
equation for the reserve is the only one he did not (care to?) publish
and, yet, it is the only one to which his name is attached, 
thus earning him lasting fame.
He showed the differential equation to
colleagues in 1875, but it
was not made public until Gram (1910) discussed it in an obituary on
Thiele. It was only in 1913 that the result appeared in a scientific text
(J{\o}rgensen, 1913) and was properly attributed to its inventor.

Thiele's reticent personality, carefully portrayed by Gram (1910) (see
Johnson and Kotz (1997) for a translated excerpt), may help to explain
why it was left to others to disseminate, develop, and also rediscover this
and other fundamental ideas he had and why his reputation does not
adequately reflect his influence.
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\noindent
{\bf References}

\noindent
Berger, A. (1939). {\em Mathematik der Lebensversicherung}. Verlag von Julius
Springer, Wien.

\noindent
Gram, J.P. (1910). Professor Thiele som aktuar. {\em Dansk
Forsikrings{\aa}rbog
1910}, 26-37.

\noindent
Hald, A. (1981). T.N. Thiele's contributions to statistics.
{\it Internatonal Statistical Review}, {\bf 49}, 1-20.

\noindent
Hald, A. (1998). Preprint No. 2. Dept. of Theoretical Statistics. Univ. of
Copenhagen.

\noindent
Hald, A. (1998). {\em A History of Mathematical Statistics from 1750 to
1930}.
Wiley, New York.

\noindent
Hansen, C. (1946). {\em Om Thiele's Differentialligning for
Pr{\ae}miereserver i
Livsforsikring.} H. Hagerups Forlag, Copenhagen.

\noindent
Hoem, J.M. (1980). Who first fitted a mortality formula by least squares?
Bl\"atter der DGVM, {\bf 14}, 459-460.

\noindent
Hoem, J.M. (1983). The reticent trio: Some little-known early discoveries
in insurance mathematics by L.H.F. Oppermann, T.N. Thiele, and J.P.
Gram. {\em International Statistical Review}, {\bf 51}, 213-221.

\noindent
Hoem, J.M. (1988). The versatility of the Markov chain as a tool in the
mathematics of life insurance. {\em Transactions of the
XXIII Congress of Actuaries, Vol.
R}, 171-202.

\noindent
Johnson, N.L. and Kotz, S. (eds.) (1997). {\em Leading Personalities in
Statistical Science}. Wiley, New York.

\noindent
J{\o}rgensen, N.R. (1913). {\em Grundz\"uge einer Theorie der
Lebensversicherung.}
Fischer, Jena.

\noindent
Nielsen, N. (1912). {\em Matematiken i Danmark 1528-1928.} Gyldendalske
Boghandel Nordisk Forlag, Copenhagen.

\noindent
Norberg, R. (1991). Reserves in life and pension insurance. {\em 
Scandinavian Actuarial
Journal}, {\bf 1991}, 1-22.

\noindent
Norberg, R. (1992). Hattendorff's theorem and Thiele's differential equation
generalized. {\em Scandinavian Actuarial Journal}, {\bf 1992}, 2-14.

\noindent
Schweder, T. (1980). Scandinavian statistics, some early lines of
development. {\em Scandinavian Journal of Statistics}, {\bf 7}, 113-129.

\noindent
Thiele, T.N. (1889). Forel{\ae}sninger over Almindelig Iagttagelsesl{\ae}re.
Reitzel, Copenhagen.

\noindent
Thiele, T.N. (1903). {\em Theory of Observations}, Layton, London. (Reprinted
in {\em Annals of Mathematical Statistics}, {\bf 2}, 165-308 (1931).)
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\hfill{Ragnar Norberg}


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