Thiele, Thorvald Nicolai

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This article Thorvald Nicolai Thiele was adapted from an original article by Ragnar Norberg, which appeared in StatProb: The Encyclopedia Sponsored by Statistics and Probability Societies. The original article ([ StatProb Source], Local Files: pdf | tex) is copyrighted by the author(s), the article has been donated to Encyclopedia of Mathematics, and its further issues are under Creative Commons Attribution Share-Alike License'. All pages from StatProb are contained in the Category StatProb.

Thorvald Nicolai THIELE

b. 24 December 1838 - d. 26 September 1910

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.

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. Jörgensen, who directed him to university studies, not in pure mathematics, but in astronomy, which Jörgensen 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 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 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 (Jörgensen, 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.


[1] Berger, A. (1939). Mathematik der Lebensversicherung. Verlag von Julius Springer, Wien.
[2] Gram, J.P. (1910). Professor Thiele som aktuar. Dansk Forsikrings{\aarbog 1910}, 26-37.
[3] Hald, A. (1981). T.N. Thiele's contributions to statistics. Internatonal Statistical Review, 49, 1-20.
[4] Hald, A. (1998). Preprint No. 2. Dept. of Theoretical Statistics. Univ. of Copenhagen.
[5] Hald, A. (1998). A History of Mathematical Statistics from 1750 to 1930. Wiley, New York.
[6] Hansen, C. (1946). Om Thiele's Differentialligning for Pr{\aemiereserver i Livsforsikring.} H. Hagerups Forlag, Copenhagen.
[7] Hoem, J.M. (1980). Who first fitted a mortality formula by least squares? Blätter der DGVM, 14, 459-460.
[8] 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. International Statistical Review, 51, 213-221.
[9] Hoem, J.M. (1988). The versatility of the Markov chain as a tool in the mathematics of life insurance. Transactions of the XXIII Congress of Actuaries, Vol. R, 171-202.
[10] Johnson, N.L. and Kotz, S. (eds.) (1997). Leading Personalities in Statistical Science. Wiley, New York.
[11] Jörgensen, N.R. (1913). Grundzüge einer Theorie der Lebensversicherung. Fischer, Jena. Zbl 44.0279.06
[12] Nielsen, N. (1912). Matematiken i Danmark 1528-1928. Gyldendalske Boghandel Nordisk Forlag, Copenhagen.
[13] Norberg, R. (1991). Reserves in life and pension insurance. Scandinavian Actuarial Journal, 1991, 1-22.
[14] Norberg, R. (1992). Hattendorff's theorem and Thiele's differential equation generalized. Scandinavian Actuarial Journal, 1992, 2-14.
[15] Schweder, T. (1980). Scandinavian statistics, some early lines of development. Scandinavian Journal of Statistics, 7, 113-129.
[16] Thiele, T.N. (1889). Forel{\ae}sninger over Almindelig Iagttagelsesl{\ae}re. Reitzel, Copenhagen.
[17] Thiele, T.N. (1903). Theory of Observations, Layton, London. (Reprinted in Annals of Mathematical Statistics, 2, 165-308 (1931).)

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

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Thiele, Thorvald Nicolai. Encyclopedia of Mathematics. URL:,_Thorvald_Nicolai&oldid=54095