Notes preparer: Nitis Mukhopadhyay

The “Paper of the Month” for February 2018 is selected to be W. Hoeffding’s 1948 paper “A Class of Statistics with Asymptotically Normal Distribution” where he introduced the concept of U-statistics and built much of the heavy-duty machineries. It was published on The Annals of Mathematical Statistics Vol. 19, No. 3 (1948), pp. 293-325. It was included in the “Breakthroughs in Statistics Volume 1: Foundations and Basic Theory” which was edited by S. Kotz and N. L. Johnson (1992, Springer, New York) with a lengthy and immensely valuable introduction prepared by P. K. Sen.

I suggested W. Hoeffding’s 1948 paper from the Annals of Math Stat where he introduced the concept of U-statistics and built much of the heavy-duty machineries. It was included in the “Breakthroughs in Statistics Volume 1: Foundations and Basic Theory” which was edited by S. Kotz and N. L. Johnson (1992, Springer, New York) with a lengthy and immensely valuable introduction prepared by P. K. Sen.

This paper changed completely the face of nonparametric statistics – both theory and practice – by creating the fundamentals to take this field to maturity. The Hoeffding paper indeed impacted many areas of classical statistical inference by showing how one must handle the probability theory behind the CLT and SLLN for dependent sequences of random variables. It gave rise to new and challenging pathways to handle moderate and large-deviation theories in the case of dependent sequences of random variables.

I totally fell in love with Hoeffding’s 1948 paper in the early 70’s when I was a PhD student at the Indian Statistical Institute-Calcutta. I had great opportunities to use his fundamental projections, CLT, and SLLN with total practical relevance of martingales and reverse martingales in my 1975 thesis-work. Personally, I have gone back to Hoeffding’s paper several hundred times in my life for its fundamentally breakthrough and truly fresh ideas. Beauty in this paper remains astonishing and unmatched given its path-breaking influences on numerous big sub-fields within statistics and probability theory including nonparametrics, large-sample theory, CLT’s for dependent processes and invariance principles, Berry-Esseen rates, theory of martingales and reversemartingales, tightness, geometric probability, and CLT’s for convex bodies.

I feel lucky that I could learn to appreciate the marvel of U-statistics theory early on (beginning 1970) in my career from P. K. Sen, Malay Ghosh, and Bob Serfling (all had close ties with Hoeffding) which helped me to build a large volume of my own significant publications to show for it. I also had the great pleasure of advising a number of PhD students (including Gaute Vik, John Judge, Mabel Moreno, and Bhargab Chattopadhyay) whose research have been significantly touched and influenced by Hoeffding’s paper. Two present students’ (Jun Hu and Chen Zhang) research overwhelmingly exploit Hoeffding’s theory of U-statistics. Our own Rick Vitale (with Herman Rubin) gave a complete asymptotic theory for square-integrable U-statistics in their 1980 Annals of Stat. paper.

After nearly 70 years of publication, W. Hoeffding’s original 1948 paper still inspires me and energizes me big time. Upon rereading the original paper the n-th time, I continue to surprise myself by discovering something interesting, deep and new that remained hidden from me during my previous passes.

Happy reading of Hoeffding (1948) paper. Just live it with TLC!