## Obituary: Theophilos Cacoullos

*Submitted by C. A. Charalambides and N. Balakrishnan*

Cacoullos’s master’s thesis, with Alan Birnbaum, on median-unbiased estimation prompted his paper, “Combinatorial Derivation of the Distribution of the Truncated Poisson Sufficient Statistic,” published in the *Annals of Mathematical Statistics*. In 1989, Cacoullos characterized normality under spherical symmetry by the admissibility of Fisher’s linear discriminant function in a paper in *Statistics and Probability Letters*.

He established that, within the family of spherical normal scale mixtures, the normal maximizes the minimax probability of correct classification, provided the Mahalanobis distance between the two populations is the same.

In the early 1960s, Cacoullos worked as a research associate at Stanford University. There, he developed ties with Herman Chernoff, Ingram Olkin, and Charles Stein, among others. He worked with Olkin, producing their joint 1965 *Biometrika* paper on the bias of functions of the characteristic roots of a random matrix.

In May 1962, Cacoullos returned to Cyprus, accepting the position of the director of the department of statistics and research of the new independent Republic of Cyprus. To pursue an academic career, however, he returned to the US, accepting an assistant professorship at the University of Minnesota. Yet, during his 15-month administrative work in Cyprus, he prepared his 1966 paper on nonparametric multivariate density estimation. In Minneapolis, he worked on characterizations of normality by constant regression of quadratic and linear forms. During the same period, he discovered the interesting relation between the t and F distributions, published in *JASA* in 1965, and also exploited in relation to homoscedasticity tests for a bivariate normal.

In 1965, Cacoullos returned to New York to take a position in the department of industrial engineering and operations research at New York University. While at New York University as associate professor, he was elected chair of probability and statistics at the University of Athens and, in 1968, returned to Greece to begin his 31-year career as mathematical and physical sciences faculty.

An excellent teacher who never used notes but did use spontaneous humor, Cacoullos no doubt influenced many of his students. In addition to standard courses in statistics, he introduced combinatorics, linear programming, stochastic processes, and actuarial mathematics. In fact, he is an honorary member of the Greek Actuarial Society.

Equally as important for the future of research in mathematical statistics in Greece was Cacoullos’s supervision of several young mathematicians. C. A. Charalambides was his first doctoral student. Subsequently, many prominent colleagues earned their PhDs in mathematical statistics under his supervision or joint supervision. Among them are M. Koutras, V. Papathanasiou, and N. Papadatos, now professors of mathematical statistics.

Cacoullos’s interest in characterizations was renewed in the 1980s with characterizations of mixtures and priors by posterior expectations. With H. Papageorgiou, a fresh PhD at the time, Cacoullos coauthored several articles characterizing discrete distributions by a conditional and a regression function.

Cacoullos visited McGill University in 1972–1973 and 1975–1976, always enjoying the collaboration and friendship of H. Ruben. Later in the fall of 1983, he gave a series of special lectures at the MIT Statistics Center at Chernoff’s invitation. He visited the Center for Multivariate Analysis under the directorship of C. R. Rao, then at the University of Pittsburgh, in the spring of 1989.

Cacoullos published more than 70 papers in international journals. Recently, he became interested in the nature and history of probability and statistics. He published seven textbooks on probability and statistics in Greek and edited *Discriminant Analysis and Applications*, published in 1973 by Academic Press. In 1989, Springer published his book *Exercises in Probability*. His recent contributions on Chernoff-type and Cacoullos-type variance generated much interest and produced a variety of publications.