Abstract
Use of the delta method in statistics and econometrics is ubiquitous. Its mention can be found in almost all advanced statistics and econometrics textbooks but mostly without any reference. It appears that nobody knows for certain when the first paper on the topic was published or how the idea was first conceived. A seemingly unrelated method to find the asymptotic variance of a statistic involving one or more nuisance parameters was given by Pierce (Ann. Stat 10, 475–478 1982). In the first part of the paper a comprehensive review of the delta method is presented with the objective of unearthing its history. In the second part a comparative analytic study of the delta method with the Pierce method is presented.
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Notes
Charles Spearman (1863-1945) was a British Psychologist known for his contributions in statistics, mostly for his rank correlation coefficient called Spearman’s rank correlation coefficient.
Karl J. Holzinger (1892-1954) was an American educational psychologist known for his work in factor analysis and in psychometrics, such as, developing models for human intelligence and cognitive test scores.
Joe Doob (1910-2004) was an American mathematician specialising in stochastic processes. He served as a professor at the Department of Mathematics in University of Illinois at Urbana Champaign from 1935 until his retirement in 1978. He is most popularly known for Doob martingale.
Robert Dorfman (1916-2002) was an American professor of political economy who made great contributions to the field of economics and statistics. He is more popularly known for his 1958 book Linear Programming and Economic Analysis with Paul A. Samuelson and Robert M. Solow.
Sewall Green Wright(1889-1988) was an American geneticist who is popularly known for his work on evolutionary theory and path analysis.
Since readers rarely have an access to the inner workings of academic research and its eventual publication, it will not be out of place to recount that story here how Professor Jay M. Ver Hoef found a copy of Dorfman (1938). Ver Hoef first learnt about the delta method as a graduate student at the Iowa State University (ISU) in the late 1980’s from Professor William Q. Meeker, who directed him to Dorfman’s paper on the delta method in the obscure journal Biometric Bulletin. In the ISU library, Ver Hoef found all four volumes of the journal, and he carried his copy of Dorfman’s paper for years with him until he realized Dorfman was almost never cited, so he wrote a manuscript crediting Dorfman. Possibly given the confident title “Who Invented the Delta Method” of Ver Hoef (2012), the then editor of the American Statistician, Professor Ronald Christensen warned him about the possibility of earlier references of the delta method. Sure enough, soon after its publication, Professor Stephen Portnoy of the University of Illinois at Urbana-Champaign (UIUC) wrote his comment, Portnoy and Ver Hoef (2012), pointing out the contribution of Doob (1935).
This seems to be a characteristic of Cramér’s book which contains many original results. For instance, Cramér (1946, p. 480) states and proves some of the most important results in statistics, such as the Cramér-Rao lower bound (CRLB). Most interestingly as we will briefly discuss in Section 2.6, there is a connection between the delta method and the CRLB.
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Acknowledgments
We are most grateful to Professor Jay M. Ver Hoef for providing us an electronic copy of Dorfman (1938) which is not available online and giving many pertinent comments and suggestions that helped us in revising the paper. Thanks are also due to an anonymous referee for pointing out mistakes in the earlier version of the paper and suggesting several improvements. We are very thankful to Professor Stephen Portnoy for his interest in our work and useful suggestions. An earlier version of the paper was presented at the Department of Statistics, University of Illinois at Urbana-Champaign (UIUC). We would like to thank the participants of the seminar for their enthusiastic reception of our work and pertinent comments. Thanks are also due to Professors Eun Yi Chung, Osman Doğan and Subrata Kundu for helpful comments. However, we are solely responsible for any remaining shortcomings.
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Osip Mandelstam (1891-1938), Russian poet and essayist. “One cannot launch a new history- the idea is altogether unthinkable; there would not be the continuity and tradition. Tradition cannot be contrived or learned. In its absence one has, at the best, not history but “progress”- the mechanical movement of a clock hand, not the sacred succession of interlinked events.”
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Bera, A.K., Koley, M. A History of the Delta Method and Some New Results. Sankhya B 85, 272–306 (2023). https://doi.org/10.1007/s13571-023-00305-9
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DOI: https://doi.org/10.1007/s13571-023-00305-9