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\noindent{\bf Harold HOTELLING} \\
b. 29 September 1895 - d. 26 December 1973
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\noindent
{\bf Summary.}  A major developer of the foundations of statistics and an important 
contributor to mathematical economics, Hotelling introduced the
multivariate $T^2$, principal components analysis, and canomical
correlations.
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Harold Hotelling was born in Fulda, Minnesota, USA.  He graduated
with a B.A. in Journalism in 1919, 
and subsequently obtained a  Master of Science degree in mathematics in 1921, both from the
University of Washington.  In 1924, he received his doctorate in mathematics from
Princeton University, writing a dissertation on topology.  Upon
leaving Princeton, he joined the Food
Research Institute at Stanford University where he was a research associate from 
1924 to 1927, and an associate professor in the Department of Mathematics
from 1927 to 1931.  It was during this time 
that Hotelling, already involved in economics and mathematics research, began
his work in statistics.  He became aware of the
writings of R. A. Fisher (q.v.) and began a correspondence with him that led to
their long friendship.  In fact, Hotelling wrote in 1927 the book review of the first edition of
Fisher's {\it Statistical Methods for Research Workers} for {\it The Journal of the American 
Statistical Association}, and also
spent a number of months visiting Fisher at the Rothamsted Experimental Station in the 
second half of 1929.  
While at Stanford, Hotelling wrote several
ground-breaking papers in both statistics and mathematical economics. Hotelling 
developed his generalization of
Student's $t$-ratio to deal with multivariate correlated response data (Hotelling,
1931) and this test known as Hotelling's $T^2$, remains one of  his best known 
results. Also he worked with Holbrook Working, and together in 1929 they developed 
results on the standard error of the estimated regression line leading to the well known 
Working-Hotelling curved simultaneous
confidence regions for a regression line.  
In addition to his statistical 
research at Stanford, Hotelling worked in mathematical economics.
 Of the notable papers he wrote, one in 1929 
dealt with a problem of optimizing price and location in a competition between two entities in a 
spatial setting.  The other
paper in 1931 dealt with exhaustible natural resources, in
which he showed that in equilibrium, 
the prices of such natural resources 
 will have a tendency  to rise over time at a percentage rate that equals the national interest rate.  

In 1931, Hotelling was recruited to Columbia University to be a professor 
of economics, and  also to undertake the initiation of
mathematical statistics at Columbia.  On July 1, 1942, Hotelling, W. Allen 
Wallis and Jacob Wolfowitz became the charter
members of the renowned Statistical Research Group (SRG) 
which was  based at Columbia University during World War II and remained  
in existence until September 30, 1945.  
(See Wallis
(1980) for details concerning the SRG.) 
 The SRG attracted an extraordinary group of
research statisticians to Columbia, and its goal was to support the Armed Forces in improving 
the quality and efficency of their war
efforts.  As part of the SRG effort, Hotelling developed methods for
control charts for multivariate data (Hotelling, 1947), and Abraham Wald
did his fundamental research  establishing the field of sequential analysis.

During his career at Columbia University  prior to the War, Hotelling continued with his innovative 
developments in statistics and
mathematical economics.  He introduced the idea of principal component 
analysis in his 1933 and 1936 papers as a way of understanding the structure of large numbers of
correlated multivariate observations, and he generalized 
the notions of correlation and multiple correlation to introduce canonical
correlation analysis which allows one to measure the strength of relationship 
between two dependent sets of multivariate
observations, and also understand the structure of the relationships between 
them
(Hotelling, 1936).  In 1940, he wrote with great foresight a paper on 
how statistics should be best taught in universities.  Hotelling (1940) anticipated  the growth of 
statistics and, at that
time, innovatively argued that the subject of statistics should be taught in universities by 
statisticians in departments of
statistics. In mathematical economics, two notable papers of this
period were one in 1935 uniting demand and utility and one in 1938
introducing the ``welfare equilibrium principle.'' 

Hotelling was president of the Institute of Mathematical Statistics in 1941, having been one of 
the three original
founding leaders of the Institute in 1935.  He also served as president of the Econometrics 
Society in 1936-37, and was actively
involved with the Cowles Commission essentially from its beginning.  

One year after the end of the World War II, Hotelling was recruited in
1946 to
the University of North Carolina at Chapel Hill to start upon his arrival a Department of 
Mathematical Statistics.  This department was to be a complement to
the Department of Experimental Statistics at North Carolina State University at Raleigh, and 
together the two departments plus groups in social sciences,
sociology, and biostatistics were to
constitute an Institute of Statistics.  Hotelling remained chairman at Chapel Hill until 1952 
and formally retired in 1966;
however, aftewards he continued to remain active in the department.  At 
North Carolina, he continued his research,  publishing papers in 1947
and 1951 on hypothesis
tests related to multivariate analysis of variance, and introduced the criteria which 
continues to be referred to as the Lawley-Hotelling trace.  

At both Columbia University and at the University of North Carolina, Hotelling 
excelled at attracting outstanding colleagues.    
Hotelling was intimately involved in
bringing Abraham Wald to Columbia, as well as helping to attract Jacob Wolfowitz and W. Allen 
Wallis.  At Chapel Hill such
renowned statisticians as R. C. Bose, Wassily Hoeffding, P. L. Hsu, William Madow, Herbert Robbins, 
and S. N. Roy, were drawn there by Hotelling.
He also was successful at developing young researchers who later would be renowned in their careers.  
He was an early mentor of both Milton
Friedman and Kenneth Arrow, both of whom were later to win the Nobel Prize in Economics.  

Hotelling's best known statistical contributions in multivariate
analysis remain vital and current to present times. These specifically
include Hotelling's $T^2$, principal
components, and a number of correlational techniques.


Although introduced in 1908, Student's $t$-test remains as one of the
most used statistical procedures for univariate data.  Hotelling (1931) had recognized that
experiments often have multiple measurements on each individual, and
thus multiple univariate $t$-tests would be correlated.  Hotelling's
elegant solution was to propose a vector formulation of Student's test
which yields a quadratic form whose distribution under the null
hypothesis is that of an $F$-distribution.  In particular, he showed
that the general distribution of the $T^2$ does not depend on nuisance
parameters, but only on a quadratic form in the population mean vector,
in which the matrix of the quadratic form is the inverse of the population
covariance matrix.  In showing this, Hotelling made use of invariance,
thereby anticipating a theory developed much later.  The multivariate
version of Student's $t$-statistic remains known as Hotelling's $T^2$
statistic.

To untangle the correlation and variability that exist among multiple
measurements $x_1 , \ldots x_q$ on an individual, Hotelling (1933)
introduced
principal components.  Principal components are uncorrelated linear
combinations of the original measurements, each successfully decreasing
in variation, and yet in total preserving the variation of the original
measurements.  Hotelling showed that the theoretical solution to the
principal component problem involved finding the characteristic roots of
a population covariance matrix.  This then naturally led to the study of
the distribution of the roots of the sample covariance matrix and thereby
opened a new research area of statistics involving roots of determinantal
equations.  Because characteristic roots of a covariance matrix were
not readily computed numerically at that time, Hotelling suggested a
power method which had the effect of accentuating the largest and
smallest roots.  This procedure was adopted for sometime by numerical
analysts until more effective factorization methods were later
developed.

Beginning with his early research, Hotelling was interested in
relations between variables.  His first publication in 1925 dealt with
the distribution of correlation ratios.  In 1936, he introduced the
concept of canonical correlations.  To handle relationships 
between two
sets of multiple measurements on an individual, he extended the multiple
correlation coefficient which had been previously introduced 
 to study the correlation between a single variable $y$ and
multiple measures $x_1 , \ldots , x_q$.  The first canonical correlation between
measures $y_1 , \ldots , y_p$ and $x_1 , \ldots , x_q$ is the maximum
correlation between separate normalized linear combinations of the x's and the
y's.  Subsequent canonical correlations are similarly defined, but
constrained to be uncorrelated of earlier chosen canonical variates.

In data analysis, the first canonical variate provides what
might be termed ``the most predictable criterion,'' which was studied by
Hotelling in 1935 in the context of educational psychology.
In 1936, Hotelling, jointly with Margaret Pabst, studied rank
correlations, and in 1953 presented a Royal Statistical Society paper on
correlations and their transforms.  This paper remains as one of the definitive
papers that discusses properties of correlation and Fisher's
$z$-transformation on Fisher's $z$-transformation.

For other reviews and discussions about Hotelling, see Olkin, Ghurye,
Hoeffding, Madow and Mann (1960), Rubin (1960), and Darnell (1990).  A
complete bibliography for Hotelling is given in Olkin {\it et al}
(1960) and his articles on mathematical economics are reprinted in
Darnell (1990).

Hotelling was married in 1920 to Floy Tracy and they had two children.
She died in 1932, and Hotelling later married
Susanna Edmundson in 1934 and 
together they had five sons, and a daughter (who died in her infancy).

Hotelling was elected to the National Academy of Sciences in 1970 and received a 
number of other prestigious honors during his
life. After a stroke the preceding year, Harold Hotelling died on December 26, 1973. 
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\begin{thebibliography}{3}

\bibitem{1}  Darnell, Adrian (1990).  {\it The Collected Economics Articles of
Harold Hotelling.}  Springer Verlag, New York.

\bibitem{2}  Hotelling, H. (1931).  The generalization of Student's ratio.  {\it Annals of 
Mathematical Statistics}, 
{\bf 2}, 360-378.

\bibitem{3}  Hotelling, H. (1933).  Analysis of a complex of statistical variables into 
principal components.  
{\it Journal of Educational Psychology}, {\bf 24}, 417-441, and 498-520.

\bibitem{4}  Hotelling, H. (1936).  Relations between two sets of variates.  {\it
Biometrika}, {\bf 27}, 321-77.

\bibitem{5}  Hotelling, H. (1940).  The teaching of statistics.  {\it Annals of 
Mathematical Statistics}, {\bf 11},
457-70.

\bibitem{6}  Hotelling, H. (1947).  Multivariate quality control, illustrated by the air testing 
of sample bombsights.
In {\it Selected Techniques of Statistical Analysis}.  (Eds. C. Eisenhart, 
M. W. Hastay, and W. A. Wallis).
Chapter 3.  McGraw-Hill, NY.

\bibitem{7}  Olkin, I., Ghurye, S. G., Hoeffding, W., Madow, W. G., and Mann, H. B. (1960).
{\it Contributions to Probability and Statistics:  Essays in Honor of
Harold Hotelling}.  Stanford University Press, Stanford, CA.

\bibitem{8}  Rubin, H. (1960).  Preface to ``Three papers in honor of Harold
Hotelling at 65,'' {\it The American Statistician}, {\bf 14}, 15.

\bibitem{9}  Wallis, W. A. (1980).  The Statistical Research Group, 1942-45.  {\it Journal of the American
Statistical Association}, {\bf 75}, 320-330.
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\hfill{Ingram Olkin and Allan R. Sampson}

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