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 ...

mathematical analysis cannot derive this boundary in any satisfactoryfashion.4) Statistical inferences. Chebyshev solved two problems which,however, were considered before him. In the first of these he (pp. 187 – 192)derived the Bayes limit theorem (§ 5.2) but did not cite anyone, and in thesecond he (pp. 193 – 201) studied the probability of a subsequent result inBernoulli trials. An event occurred m times in n trials; determine theprobability that it will happen r times in k new trials. Guiding himselfmostly by the Stirling theorem, Chebyshev non-rigorously derived anintegral limit theorem similar to that obtained by Laplace (§ 7.1-5).5) Mathematical treatment of observations (pp. 224 – 252). Chebyshev (p.227) proved that the arithmetic mean was a consistent estimator of theunknown constant. Unlike Poincaré (§ 11.3-7), he (pp. 228 – 231) justifiedits optimality by noting that, among linear estimators, the mean ensured theshortest probable interval for the ensuing error. The variance of thearithmetic mean was also minimal (Ibidem); although Chebyshev had notpaid special attention to that estimator of precision, it occurred that he, inprinciple, based his reasoning on the definitive Gaussian substantiation ofthe MLSq (§ 9.1.3).At the same time Chebyshev (pp. 231 – 236) derived the normaldistribution as the universal law of error in about the same way as Gauss didin 1809. The Gauss method, Chebyshev (p. 250) maintained, bearing inmind exactly that attempt later abandoned by Gauss, was based on thedoubtful law of hypotheses, – on the Bayes theorem with equal priorprobabilities. Chebyshev several times censured that law when discussingthe Bayesian approach in his lectures and he (p. 249) wrongly thought thatthe Gauss formula (9.6b) had only appeared recently and that it assumed alarge number of observations. He did not mention that the Gauss formulaprovided an unbiassed estimation. It might be concluded that the treatmentof observations hardly interested him.6) Cancellation of a fraction (pp. 152 – 154). Determine the probability Pthat a random fraction A/B cannot be cancelled. Markov remarked thatKronecker (1894, Lecture 24) had solved the same problem and indicatedDirichlet’s priority. Kronecker had not supplied an exact reference and I wasunable to check his statement; he added that Dirichlet had determined theprobability sought if it existed at all. Anyway, Bernstein (1928/1964, p. 219)refuted Chebyshev’s solution and indicated (p. 220), that the theory ofnumbers dealt with regular number sequences whose limiting or asymptoticfrequencies of numbers of some class, unlike probabilities, which we willnever determine experimentally, might be studied. See Postnikov (1974) onthe same problem and on the stochastic theory of numbers.12.3. Some General ConsiderationsAnd so, Chebyshev argued that the propositions of the theory ofprobability ought to be rigorously demonstrated and its limit theoremsshould be supplemented by estimation of the errors of pre-limiting relations(Kolmogorov 1947, p. 56). He himself essentially developed the LLN and,somewhat imperfectly, proved for the first time the CLT; on the study ofthese two issues depended the destiny of the theory of probability (Bernstein126

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