Pranab Kumar Sen

Pranab Kumar Sen (7 November 1937 – 31 December 2023) was an Indian-American statistician whose research concerned nonparametric statistics, rank-based inference, sequential methods, and statistical applications in biomedical research. He spent most of his academic career at the University of North Carolina at Chapel Hill, where he held the Cary C. Boshamer Professorship of Biostatistics. His work connected the finite-sample properties of rank statistics with asymptotic theory and extended those methods to multivariate, censored, and sequentially observed data.

Early life and education

Sen was born in Calcutta, then part of the Bengal Presidency in British India. He studied statistics at the University of Calcutta, receiving a bachelor's degree in 1955 and a master's degree in 1957. His education coincided with the institutional development of modern statistics in India, particularly through the University of Calcutta and the Indian Statistical Institute.

He completed his doctorate at the University of Calcutta in 1962 under the supervision of Hari Kinkar Nandi. His doctoral research examined distribution-free procedures based on ranks and order relations. These methods replaced assumptions about a population's complete probability distribution with inference derived from the relative positions of observations.

Sen subsequently worked at the Indian Statistical Institute before moving to the United States. In 1965 he joined the faculty of the University of North Carolina at Chapel Hill, where an established program in mathematical statistics was developing alongside the university's departments of biostatistics and public health.

Rank-based inference

A central part of Sen's research concerned statistics constructed from ranks rather than from the original numerical magnitudes of observations. Rank procedures retain information about ordering while reducing dependence on a specific parametric model. This property makes them applicable when the normal-distribution assumptions underlying conventional regression and analysis-of-variance methods are not appropriate.

Sen studied the asymptotic behavior of rank statistics under both null hypotheses and sequences of local alternatives. His analyses established conditions under which normalized rank statistics converge to limiting distributions suitable for hypothesis testing and interval estimation. He also investigated how dependence between observations changes those limits, an issue that became increasingly important in longitudinal studies and other forms of correlated data.

His 1968 article “Estimates of the Regression Coefficient Based on Kendall's Tau” developed a rank-based estimator for the slope in a simple linear model. The estimator takes the median of the pairwise slopes determined by distinct observations. Henri Theil had introduced a closely related construction in 1950, and Sen's formulation extended its statistical interpretation and treatment of ties. The resulting procedure is generally called the Theil–Sen estimator.

The estimator is invariant under monotone transformations that preserve the relevant ordering, and it is less sensitive to isolated extreme observations than the ordinary least-squares estimator. Its theoretical connection to Kendall's rank correlation coefficient places regression estimation within the broader framework of concordance between ordered pairs.

During the late 1960s, Sen's group at Chapel Hill also compared asymptotic rank procedures with exact finite-sample calculations. Research associate You Watanabe contributed permutation calculations for the regression-rank statistics considered in this work. Those calculations supplied finite-sample reference distributions against which the limiting approximations were evaluated, particularly in configurations containing tied observations or unevenly spaced design points.

Multivariate and sequential methods

Sen extended rank theory beyond one-dimensional samples by studying multivariate observations whose components could not be reduced to a single natural ordering. This required statistics assembled from componentwise ranks, spatial comparisons, or combinations of marginal scores. The resulting methods preserved distribution-free features under specified null models while accounting for dependence among the measured variables.

With Madan Lal Puri, Sen wrote Nonparametric Methods in Multivariate Analysis, published in 1971. The book organized contemporary work on multivariate rank tests and presented asymptotic methods for evaluating their joint behavior. It became part of the mathematical literature connecting classical nonparametric inference with later developments in robust and semiparametric statistics.

Another sustained area of his research was sequential analysis, in which the amount of data is not necessarily fixed before observation begins. Sen examined rank procedures under stopping rules and studied the consequences of repeatedly evaluating accumulating evidence. This work addressed the distinction between a statistic's behavior at a predetermined sample size and its behavior when the sample size is itself random.

In collaboration with Malay Ghosh and Nitis Mukhopadhyay, Sen later developed a systematic treatment of sequential estimation. Their work considered estimation schemes in which sampling continued until a prescribed accuracy criterion was attained. The analysis combined stopping-time arguments with asymptotic expansions that described the additional observations required when unknown population quantities had to be estimated during sampling.

Biostatistical applications

Sen's appointment in biostatistics shaped the application of his theoretical research. Biomedical data frequently include censoring, repeated measurements, incomplete observations, and response distributions that differ from standard parametric models. Rank procedures provide a common mathematical structure for several of these settings, although the effects of censoring and dependence must be incorporated into their variance calculations.

His work on survival analysis examined rank statistics constructed from event times when some subjects leave observation before the event occurs. In such data, the observable ordering is only partial because a censored survival time does not reveal the subject's eventual event time. Sen analyzed weighted rank procedures that used the available risk-set information without treating censored values as fully observed outcomes.

He also studied repeated-measures and longitudinal designs, in which measurements from the same individual are correlated. His asymptotic formulations separated variation attributable to individual observations from variation induced by clustering. This distinction contributed to the use of rank-based techniques in medical trials and epidemiological studies where a fully specified likelihood was unavailable or undesirable.

The relation between mathematical assumptions and biomedical sampling structures remained a recurring feature of his work. Rather than treating distribution-free inference as the absence of assumptions, Sen formulated the exchangeability, independence, continuity, and censoring conditions under which rank procedures retained their stated properties.

Academic career and publications

Sen remained at the University of North Carolina for the principal part of his career and later became professor emeritus. He supervised doctoral research across mathematical statistics and biostatistics while participating in the university's probability and statistics programs. His publications included research articles, edited collections, and monographs addressing nonparametric inference, sequential estimation, asymptotic theory, and biomedical applications.

Madan Lal Puri's collaborations with Sen concentrated on multivariate nonparametric methods and the systematic presentation of rank-based inference. Malay Ghosh and Nitis Mukhopadhyay worked with him on sequential estimation, particularly on stopping rules designed around precision requirements. These collaborations reflected the movement of Sen's research between general limit theory and statistical problems defined by specific sampling arrangements.

Sen served the professional organizations associated with his discipline and was elected a fellow of the Institute of Mathematical Statistics and the American Statistical Association. He was also an elected member of the International Statistical Institute. In 2010, the American Statistical Association awarded him the Samuel S. Wilks Memorial Award for contributions to statistical methodology and its applications.

Place in statistical theory

Sen's research formed part of the postwar development of nonparametric statistics from a collection of individual tests into a general field of inference. His work emphasized that rank procedures could support estimation, multivariate comparison, sequential decision rules, and analysis under censoring rather than being confined to elementary two-sample testing.

The Theil–Sen estimator remains the most widely cited construction associated with his name. Its continued use reflects the broader role of median-based pairwise comparisons in robust statistics. His other work contributed to the mathematical foundations through which such procedures are evaluated, including consistency, asymptotic normality, efficiency under local alternatives, and behavior under random stopping.

Sen died on 31 December 2023. The University of North Carolina's statistical community subsequently continued the P. K. Sen Distinguished Lecture series, which had been established in connection with his contributions to mathematical statistics and biostatistics.