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\noindent{\bf Carlo Emilio BONFERRONI}\\
b. 28 January 1892 - d. 18 August 1960
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\noindent{\bf Summary.} His name is attached to the Bonferroni Inequalities which
facilitate the treatment of statistical dependence. Improvements to the inequalities
have generated a large literature.

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Carlo Emilio Bonferroni was born in 1892 at Bergamo,
a university town in northern Italy.
He studied conducting and piano at the conservatory
in Torino (Turin), and then
studied for the degree of {\it laurea}
in mathematics
in Torino under Peano and Segre.
He spent a year broadening his education
in Wien (Vienna) at the University, and
in Z\"urich at the Eidgen\"ossicher Technischen
Hochschule.
During the 1914-1918 war he served as
an officer in the engineers.
He became
{\it incaricato} (assistant professor) at the Turin Polytechnic,
and then in 1923 took up the chair  of financial mathematics
at the Economics Institute in Bari
where he was also Rector for 7 years.
In 1933 he transferred to Firenze (Florence) where he held his chair
until his death in 1960.
He was Dean of his Faculty for five years.
He received honours from within his own country, but the only one
from outside was from the Hungarian Statistical Society.

The obituary of him by
Pagni
lists his works under three
main headings: actuarial mathematics (16 articles, 1 book); 
probability and statistical
mathematics (30, 1); analysis, geometry and rational mechanics (13, 
0).
His name is familiar in the statistical world through
his 1936 paper in which the Bonferroni Inequalities first appear.
If $p_i$ is the probability
of having characteristic $i$,
$p_{ij}$ the probability of having
$i$ and $j$  and so on then
he introduces the notation
$S_0 = 1, \quad S_1 = \sum p_i, \quad S_2 = \sum p_{ij}, 
\quad S_3 = \sum p_{ijh},$ ...
Then writing $P_r$ for the probability of
exactly $r$ events 

\[
P_0 \leq 1 , P_0 \geq 1 - S_1,
P_0 \leq 1 - S_1 + S_2,
P_0 \geq 1 - S_1 + S_2 - S_3
\]

\noindent and so on. The first of these inequalities 
is due to Boole (q.v.) in 1854.  It is the
highlighting of Boole's Inequality by Francesco Paolo Cantelli (1875-1966) as a 
tool for treating
statistical dependence at the International Congress of Mathematicians held 
at Bologna in 1928, which Bonferroni
attended, that may have led Bonferroni to produce the elegant pattern of the 
other members of the
sequence. Attribution to Boole is in fact made on pp. 4 and 25 of Bonferroni's 
paper , in which the
inequalities are justified in ``symbolic" fashion". Similar ideas using $S_k$'s 
were pursued by
K\'aroly Jordan (q.v.) and Henri Poincar\'e. 

The inequalities achieved their popularity
through a book of Maurice Fr\'echet (q.v), a frequent correspondent of 
Cantelli's, of 1940;
and William Feller's celebrated {\it An Introduction to Probability 
Theory and its Applications,}, Vol.1,
first published in 1950,
which was probably the source for most English speaking readers,
It is interesting to note that he cites only the monograph
by Fr\'{e}chet.

The inequalities have given rise to a large
literature. The first two have been used in particular
in simultaneous statistical inference.
The method known as Bonferroni adjustment
usually relies only on Boole's Inequality.


Bonferroni's 1936 article uses the classical definition of probability, in terms of
a finite sample space of equally likely events, usually attributed to Laplace (q.v.).
His notion of probability was not, however, confined to this.
In his inaugural address for the academic year (1924-25)
published in 1927 he clearly states:


\begin{quote}
A weight is determined directly by a balance.
And a probability, how is that determined?
What is, so to say, the probability balance?
It is the study of frequencies which gives rise to a
specific probability (p. 32)
\end{quote}

\noindent Then he moves on to consider long run frequency
in more detail.
On p. 35 he specifically denies that subjective probability
is amenable to mathematical analysis.


After these papers  he  moved away
from writing on the foundations of probability.
A reason for this change of direction could have been the
appearance of the work of
von Mises (q.v.)
now often regarded
as a landmark in the development of the frequentist
view.

Bonferroni also worked in a number of other statistical
areas.  A competitor to the
well-known Gini (q.v.) index of concentration,
Bonferroni's concentration index is designed to
measure income inequality.
Let $x_{(i)}$ be the observed $i$th order statistic
in a sample of size $n$, so that $x_{(i-1)} \le x_{(i)}
\quad(i=2, \dots, n)$.
Define
$m_i, i=1,2,...,n$, as the sample partial means, so that $m=m_n$ is the ordinary 
sample mean, by:

\[
m_i = \frac{1}{i}\sum_{j=1}^{i}x_{(j)}, \ i=1,2,...n.
\]

\noindent Then Bonferroni's index $B_n$ is

\[
B_n = 1 - \frac{1}{n-1}\sum_{i=1}^{n-1}\frac{m_i}{m}
\]

\noindent He was also interested in ``algebraic" means
which he treats extensively in his
textbook, {\it Elementi di Statistica
Generale}.
The algebraic mean $M_p$ of order $p$
is $\sqrt[p]{\frac{x_1^p + \dots + x_2^p}
{n}}$.
He published on the properties of a
generalisation of this
$M_{p+q}$ defined as
$\sqrt[p+q]
{\frac{x_1^p x_2^q + x_1^q x_2^p + \dots}
{n(n-1)}}$
and similarly for higher orders.

One reason for his current lack of recognition may be the fact that
his books were never properly
disseminated.
Apart from {\it Elementi di analisi matematica},
and that only
in its last edition of 1957, and a smaller research monograph: {\it Sulla 
correlazione e sulla connessione} (1942), they
probably do not exist in typeset versions.
One of the volumes, {\it Elementi di Statistica Generale}
was reprinted in facsimile after his death at the instigation of
the Faculty of Economics of University
of Firenze; bound with it is a memoir by de Finetti.
The reason his books were never properly
typeset is that he believed that books were too expensive for students to buy,
and so to hold costs down
he handwrote his teaching material, and had the books
printed from that version. (This pattern is reasserting itself
for the same reasons with the aid of electronic publishing.)
They run to hundreds of pages, neat and
almost correction free.
His articles have a clear explanatory nature.
He was someone with a genuine interest in
communicating his ideas to his audience.
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\noindent{\bf References}

\noindent
Benedetti, C. (1982). Carlo Emilio Bonferroni (1892-1960). {\it Metron}, 
{\bf 40}, N.3-4, 1-36.

\noindent
Bonferroni, C. E. (1936).
Teoria statistica delle classi e calcolo delle probabilit\`{a}.
{\it Pubblicazioni del R Istituto Superiore di Scienze Economiche e
  Commerciali di Firenze}, {\bf 8}, 1-62.

\noindent
Bonferroni, C. E. (1927).
Teoria e probabilit\`a.
In {\it Annuario del R Istituto Superiore di Scienze Economiche
  e Commerciali di Bari per L'anno Accademico 1925-1926}, pp. 
15-46.
  
\noindent
Bonferroni, C. E. (1941).
{\it Elementi di Statistica Generale}
Universit\`a Bocconi, Milano
(First edition 1927-28).

\noindent
Galambos, J. and Simonelli, I. (1996).
{\it Bonferroni-type Inequalities with Applications}.
Springer-Verlag, New York.

\noindent
Pagni P (1960).
Carlo Emilio Bonferroni.
{\it Bollettino dell'Unione Matematica Italiana},
{\bf 15}, 570--574.

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