1 Oscar Sheynin History of Statistics Berlin, 2012 ISBN 978-3 ...

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1 Oscar Sheynin History of Statistics Berlin, 2012 ISBN 978-3 ...

5) The MLSq (Bienaymé 1852; HS, pp. 66 – 71). Bienaymé correctlyremarked that least variance for each estimator separately was not asimportant as the minimal simultaneous confidence interval for all theestimators (as joint efficiency!). He assumed that the distribution of theobservational errors was known, made use of its first moments and evenintroduced the first four cumulants and the multivariate Gram – Charlierseries (Bru 1991, p. 13; Hald 2002, pp. 8 – 9). He determined thatconfidence interval by applying the principle of maximum likelihood,introducing the characteristic function of the vector of the errors and makinguse of the inversion formula. True, he restricted his choice of the confidenceregion, but derived here the χ 2 distribution.6) A branching process (Bienaymé 1845; HS, pp. 117 – 120). Bienayméhad formulated the properties of criticality of a branching process whileexamining the problem of the extinction of noble families that becameattributed to Galton. D. G. Kendall (1975) reconstructed Bienaymé’s proofand reprinted his note.7) When investigating the stability of statistical frequencies (see also item2 above), Bienaymé (1840a; HS, pp. 108 – 110) expressed ideas thatunderlie the notion of sufficient estimators.10.3. CournotHe intended his main contribution (1843) for a broader circle of readers.However, almost completely declining the use of formulas, he hardlyachieved his goal. Recall also (§ 8.1) that Cournot passed over in silence theLLN. I describe his work as a whole; when referring to his main book, Imention only the appropriate sections. Chuprov (1925a/1926, p. 227) calledCournot the real founder of the modern philosophy of statistics. This seemsto be exaggerated. He did not substantiate and canonically prove the LLN(Chuprov 1905/1960, p. 60; 1909/1959, pp. 166 – 168), did not evenformulate that law.1) The aim of the theory of probability. It was The creation of methods forassigning quantitative values to probabilities (p. 181). He thus moved awayfrom Laplace (§ 7.1) who had seen the theory as a means for revealing thelaws of nature.2) The probability of an event. Cournot’s definition (§ 18) includedgeometric probability, which had been lacking any formula, and combined itwith the classical case. He (§ 113) also introduced probabilities unyieldingto measurement and (§§ 233 and 240.8) called them philosophical. Theymight be related to expert estimates whose treatment is now included in theprovince of mathematical statistics.3) The term médiane. This is due to Cournot (§ 34).4) The notion of randomness. Cournot (§ 40) repeated its ancientconnection with the intersection of chains of events (my § 1.1), and, in § 43,indirectly connected randomness with unstable equilibrium by remarkingthat a right circular cone, when stood on its vertex, fell in a randomdirection. Cournot (1851, § 33, Note 38; 1861, § 61, pp. 65 – 66) alsorecalled Lambert’s attempt to study randomness (see my § 6.1.3), and(1875/1979, pp. 177 – 179) applied Bienaymé’s test (§ 10.2-4) forinvestigating whether the digits of the number π were random butreasonably abstained from a final conclusion.91

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