## The law of error. Offprint from: Transactions of the Cambridge Philosophical Society, vol. 20, part I.

[Cambridge: University Press, 1905].

First edition, extremely rare offprint, **inscribed by Edgeworth**, of his most important contribution to mathematical statistics. “The most important of the papers relating to this subject” (Bowley). “‘Of all the great economists in this book, he [Edgeworth] is (apart from Bernoulli and Slutsky) the only one to have made original contributions to mathematical statistics’ (Blaug, *Great Economists before Keynes* (1986), pp. 69-71). “In 1883 began the series of papers that were to make [Edgeworth] the leading theorist of mathematical statistics of the latter half of the 19th century” (Stigler, p. 98). “He set himself to do at last what had been talked about and assumed possible for over a century, but had never been accomplished: adapt the statistical methods of the theory of errors to the quantification of uncertainty in the social, particularly economic, sciences. In this he succeeded brilliantly” (*ODNB*). “He is most frequently remembered today for his work on the Edgeworth Series [introduced in the offered paper], but in fact he touched on nearly every sphere of modern statistics … In the 1890s, Edgeworth’s statistical work became increasingly occupied by a competition with Karl Pearson as to who could best model skew data … Edgeworth at one time or another tried three different approaches … The third was based upon what we now call Edgeworth series. The essence of Edgeworth’s approach was to generalize the central limit theorem by the inclusion of correction terms, terms that appeared in the derivation of the distribution of sums [of random variables] but which became negligible if the number of terms in the sum was large. The idea was that skew distributions found in nature were skew because they were aggregates of relatively small numbers of non-normal components. Edgeworth was thus taking a theoretical approach, one that he felt was more appealing than Pearson’s more ad hoc approach. The Edgeworth Series was foreshadowed in his work as early as 1883, but the full development came later (Edgeworth 1905 [the offered paper]) … later statisticians have found that Edgeworth’s mode of arranging correction terms was far superior to alternatives proposed by Bruns, Gram and Charlier, and the Edgeworth Series has become an important technique for approximating sampling distributions … Edgeworth was an independent thinker upon statistical matters, though he was perhaps the earliest to appreciate and follow up on Galton’s innovative concepts of regression and correlation. Edgeworth’s most important influence was upon Karl Pearson, though Pearson was chary in his recognition of this influence. Taken together, Galton, Edgeworth and Pearson shaped modern statistics to a greater degree than any other individual or group before R. A. Fisher” (Stigler, pp. 98-99). “The ‘Edgeworth series’ expansion and the ‘Edgeworth box’ diagram are the two instances in which his name is still explicitly honoured” (Barbé, p. 193). ‘The law of error’ was published in the *Cambridge Philosophical Transactions*, vol. 20, in two parts (pp. 36-65 & 113-141). The present offprint contains both parts (retaining the journal pagination), and also a 14-page ‘Continuation of Appendix VII’ which was **not published** in the *Transactions* (pp. 131-141 of the main work is an Appendix divided into seven sections). OCLC lists one copy in US (Michigan) and two copies in UK (Exeter, Oxford); COPAC adds Cambridge. There are no copies of this offprint in auction records, nor of any book or article inscribed by Edgeworth.

*Provenance*: Carl Vilhelm Ludvig Charlier, Swedish astronomer (1862-1934) (inscribed by Edgeworth on front free endpaper: ‘Professor C V Charlier // With the author’s compliments’); several pencil annotations in text (by Charlier?). Charlier “was in a more substantial manner occupied with probability and statistics only in a relatively late period of his work. Initially, his scientific activities were in perturbation theory, photometry, and scientific photography … In 1905, Charlier published the article ‘Über das Fehlergesetz’ (‘On the Law of Error’), in which he tried to demonstrate that the representation of ‘arbitrary frequency curves’ by series expansions in derivatives of a normal density function ‘could be followed from the Laplacian theory of errors in an un-coerced way’ … Only by contributions of Charlier and Edgeworth in 1905 and later years … was the connection between analytic devices in the realm of the [central limit theorem] on the one hand, and series expansions on the other, perceived by a broader audience” (Fischer, pp. 118-119).

“Francis Ysidro Edgeworth (1845-1926) was perhaps the statistician with the greatest mathematical abilities at the end of the 19th century. Later on, however, he became prominent due to his work on economics rather than his statistical contributions … Edgeworth, who had originally studied ancient languages and law, acquired his mathematical skills as an autodidact mainly by reading the works of ‘classical’ authors like Laplace, Poisson, and Fourier. He did not become familiar with the analytical development of the beginning modern era. Regarding his stochastic concepts and analytic methods, Edgeworth was especially influenced by apparently very carefully reading Laplace’s [*Théorie analytique des Probabilités*, 1812]. The latter’s remarks on possible alternatives to the method of least squares for parameter estimation were adopted by Edgeworth and further advanced, in particular in context with his frequent discussions of non-normal distributions. As a consequence of his autodidactic education, Edgeworth cultivated an analytic style which reminds one of the 18^{th} rather than the early 20^{th} century. His presentation of mathematical issues was often sketchy and not always straightforward. Altogether, these circumstances make his statistical work quite difficult to read. On the other hand, his permanent readiness to advance the discussion of statistical models far beyond their momentary practical applicability give his contributions a very modern touch.

“Within Edgeworth’s statistical work, the problem of the characteristic properties of the ‘Law of Error’ was especially prominent. The designation ‘Law of Error’ was mainly used by Edgeworth for ‘frequency laws’ which expressed the probability distribution of a random variable resulting from the coaction of several ‘elements.’ Those ‘elements’ were essentially independent elementary errors whose sum obeyed an approximate normal distribution, at least roughly. According to Edgeworth, an error law of this kind had ‘the advantage of being based on a *vera causa*, perhaps the most universal law of nature.’ And therefore, as Edgeworth pointed out again and again, these error laws could be assessed regarding their use for representing empirical frequencies in two ways: Firstly by an ‘a priori consideration’ of the plausibility of a possible mechanism based on elementary errors, by which the statistical quantity considered could be produced; secondly by testing whether the law of error (after an appropriate fitting of its parameters) could actually provide a sufficiently exact representation of the distribution of the sampling data …

“What was the nature of the most general error law that could be based on the hypothesis of essentially independent and additively coacting ‘elements’? In the case of an ‘infinite’ number of elementary errors one could expect a Gaussian distribution, apart from ‘pathological’ exceptions. Edgeworth, however, intended to substantiate the frequent occurrences of moderate deviations from normal distributions by an appropriate hypothesis of elementary errors as well. The assumption of only a modest number of elementary errors yielded modifications of the Gaussian distribution through series, which could represent ‘The Law of Error’ quite accurately. In the context of strongly asymmetric empirical distributions, the usual model of elementary errors could no longer be used. But even in such cases, Edgeworth tried to maintain the ‘natural’ character of his approximations to probability curves by means of his ‘method of translation’.

“In his first statistical paper, Edgeworth had already thoroughly discussed the use of correctional terms in addition to a normal density for representing general ‘facility-curves.’ In 1894 he submitted an article, in which he further elaborated the idea of modifying normal densities by series expansions, including a discussion on the adjustment of these expansions to statistical material. With the exception of a summary [‘The Asymmetrical Probability Curve,’ *Proceedings of the Royal Society of London *57, 563-568], this article remained unpublished, probably due to the rather clumsy presentation of the contents in the paper. A revised version was only printed in 1905 [as the offered work]. The core of this essay, which had apparently been written without knowledge about similar contributions by other authors, was the expansion of ‘frequency-loci’ – Edgeworth’s general designation for graphic representations of probability functions – assigned to sums of independent random variables by the now so-called ‘Edgeworth series’ …

“The 1905 article ‘The Law of Error’ crowns Edgeworth’s two-decade-long efforts concerning sums of elementary errors and simultaneously summarizes his previously achieved results on this and related topics. This voluminous paper is quite difficult to read, though by no means long-winded, and full of interesting details. ‘Edgeworth expansions’ are introduced and derived by three different methods. There is also a discussion of – in Edgeworth’s own words – ‘reproductive’ distributions (stable distributions in modern terminology), and their significance as limit laws for sums of identically distributed random variables is demonstrated. Edgeworth illustrated his series expansions by discussing particular cases, such as sums of two-valued or rectangularly distributed random variables. He also tried to generalize his results toward multidimensional errors, certain functions (not only sums) of elementary errors, or weakly dependent elementary errors. Edgeworth explicitly stated and discussed his assumptions on the properties of the elementary errors, even if in a verbose and not always entirely precise form. In his analytic methods, he followed Laplace and Poisson regarding the use of Fourier analysis, he was influenced by Crofton’s idea of partial differential equations, and he applied Karl Pearson’s method of moments. Apparently, Edgeworth neither knew modern contributions to the [central limit theorem] or the analytic theory of moments, nor was he familiar with the analytic style of post-Weierstrassian era. His 1905 article … nevertheless is impressive because of the analytic and stochastic intuition of its author, as well as due to its quest for far-reaching generalizations …

“Edgeworth’s comprehensive account also comprised the discussion of a further, in his own words [p. 54] ‘fresh,’ condition on the ‘sought’ law of error, to be ‘reproductive’. According to Edgeworth, those probability densities are called ‘reproductive’ which belong to a certain ‘family’ of functions with the following property:

‘if two or more independently fluctuating quantities [i.e., random variables] *A*, *B*, … assume different values with a frequency designated by a member of the family represented by the sought function, then *Q* a quantity formed by adding together each pair (triplet, etc.) of concurrent values presented by *A*, *B*, … will also assume different values with a frequency designated by a member of the sought family.’

“This portion of text, which is characteristic of Edgeworth’s idiosyncratic style, was far from being a precise definition of ‘reproductive.’ Only by the following analytical explanations was it clarified that a ‘family’ consisted of all ‘frequency functions’ of the form 1/*c**f*(*x/c*), where *c* was any positive number and *f *a given density function.

“Edgeworth [p. 54] also hinted at the fact that ‘frequency-curves’ of random variables which are composed of a ‘great number’ of independent identically distributed elementary errors are necessarily reproductive: If *A* and *B* are independent random variables, and both variables may be assumed to be additively composed of ‘great numbers’ *m*_{1} and *m*_{2} of elementary errors of the ‘typical sort,’ then the densities of *A *and *B* belong to the same family. *A + B* is a fortiori composed of a ‘great number’ *m*_{1} + *m*_{2} of independent identically distributed elementary errors of the ‘typical sort,’ and therefore has a frequency function from the family under consideration as well …

“In his 1905 article Edgeworth also considered several generalizations of the results described above. Edgeworth [p. 115] established, for example, his series expansion for sums of two-dimensional elementary errors, using the method of moments. Concerning his assumption of bounded elementary errors he mentioned that for the existence of a normal limiting distribution it would only be important that the moments of the single errors were finite and approximate among each other. Edgeworth recommended to substitute in the proofs unbounded elementary errors by bounded ones, such that the latter had almost the same moments as the first. With these – rather qualitative and only verbal remarks – he came close to the idea of truncated random variables as it was developed by Markov at exactly the same time.

“Very noteworthy, but quite difficult to understand due to their rather sketchy presentation, are Edgeworth’s ideas [pp. 121-126, 139-141] on generalizing additivity of elementary errors. To this aim the compound error was assumed to be a polynomial in elementary errors of a certain degree, where it was presupposed that in this polynomial the coefficients of higher order were small compared with the first ones … The characteristic feature of Edgeworth’s polynomial was that a sum of independent random variables (corresponding to the linear part of the polynomial) was augmented by additional quantities depending on these random variables, whose influence, however, was small compared with the elements of the linear part. In turn, Edgeworth [p. 126] tried to cover possible stochastic dependencies among elementary errors by polynomial models of this kind” (Fischer, pp. 122-132).

Born in Edgeworthstown, County Longford, into a leading Irish family, Edgeworth studied classics at Trinity College, Dublin and then *literae humaniores* at Oxford, graduating in 1869. He then studied law, and was called to the bar in 1877. After leaving Oxford, and while studying law, he also undertook an intense programme of self-study in mathematics. “In 1879 Edgeworth became in effect an apprentice in economic theory to his friend and neighbour William Stanley Jevons. Starting from scratch, he learned economics so well and so fast that in 1881 he published his masterpiece *Mathematical Psychics*. Although its reviewers Jevons and Marshall praised the book, neither showed any understanding of the profound ‘economical calculus’ that is its heart, a positive analysis of exchange in both market and non-market settings. In the latter, the possibility of coalitions between the various parties to the exchange poses severe and quite novel analytical difficulties, but even so Edgeworth was able to reach deep and highly non-intuitive theorems about relations between non-market and market exchange …

“For whatever reasons (and one can think of several), soon after *Mathematical Psychics* appeared Edgeworth changed course completely, and in 1883 began the series of papers that were to make him ‘the leading theorist of mathematical statistics of the latter half of the 19th century’ (Stigler, p. 98) … but, as with his economics, his achievements were consistently underrated for many years. Only after 1975 did sympathetic criticism by Stephen Stigler and others lead to proper evaluation of Edgeworth’s contributions, which culminated in 1996 in three large volumes (ed. C. R. McCann) which reprint nearly all his papers on statistics and probability” (*ODNB*).

Baccini 119; Bowley, *F. Y. Edgeworth’s Contributions to Mathematical Statistics**, 1928,* no. 47. Barbé, *Francis Ysidro Edgeworth*, 2010. Fischer, *A History of the Central Limit Theorem*, 2011. Stigler, ‘Edgeworth as statistician’, pp. 98-99 in: *The new Palgrave: a dictionary of economics, *ed. Eatwell, Milgate, & Newman, 1987.

4to (290 x 226 mm), pp. [2, blank], [1, divisional title], [36], 37-65, [1, blank], [113], 114-141, [3, blank], [i], ii-xiv. Original printed wrappers (a few small tears in spine with minor loss), uncut and partially unopened. Printed erratum slip laid in.

Item #4955

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Price:
$6,000.00
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