Abstract
This chapter provides an account of mathematical approaches that have been used in fitting models to distributions and in providing statistical acceptability tests of hypotheses. These helpful tools attempt to bridge the gap between finite samples and the infinite populations. Tools of interest in this chapter pertain to significance tests and fitting methods.
In terms of the underlying distributions needed for statistics, mathematicians in the past used binomial, Poisson, and normal (Gaussian) distributions. However, distributions have been expanded to include others such as the exponential distribution.
The theory of random probability distributions has been expanded so that indefinitely many distributions—far more than the 30, 40, or so that are familiar—can be constructed. This expansion of the theory of random probability distributions now permits investigators to postulate alternative underlying distributions with alternative solutions. This chapter stresses the multiple interpretations that can and should be derived from the use of statistics.
Coolly considered, this is a preposterous claim, which would have been universally rejected long ago, if those who made it had not successfully concealed themselves from the eyes of common sense in a maze of mathematics. (From pp. 388–389 in Keynes, John Maynard, 1921, A Treatise on Probability, London: MacMillan and Co.)
I should like to say: mathematics is a motley of techniques of proof.—And upon this is based its manifold applicability and its importance. (p. 84c in Wittgenstein, Ludwig, 1967, Remarks on the Foundations of Mathematics, Cambridge, MA: the M. I. T. Press, first published in 1956, edited by G. H. von Wright, R. Rhees, and G. E. M. Anscombe and translated by G. E. M. Anscombe)
George Cantor claimed that the essence of mathematics lies in its freedom. But mathematicians do not pick problems from the air for the pleasure of solving them. …Do I claim that everything that is not smooth is fractal? That fractals suffice to solve every problem of science? Not in the least. (From pp. 178 and 299 in Mandelbrot, Benoit B., 2012, The Fractalist: Memoir of a Scientific Maverick, New York: Pantheon Books)
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- 1.
These conflicts between K. Pearson and R. A. Fisher, on the one hand, and prominent Bayesians are discussed throughout McGrayne and Sharon Bertsch, 2011, The theory that would not die: how Bayes’ rule cracked the enigma code, hunted down Russian submarines & emerged triumphant from two centuries of controversy, New Haven: Yale University Press.
- 2.
See, for instance, pp. 6–20 in Neter, John, William Wasserman, and Michael Kutner, 1985, Applied Linear Statistical Models, Homewood, IL: Irwin.
- 3.
See McGrayne, op. cit., pp. 47, 49, and 55.
- 4.
See p. 3 in Mandelbrot, Benoit B., 1983, The Fractal Geometry of Nature, New York: W. H. Freeman and Company, originally 1977.
- 5.
See Taleb, Nassim Nicholas, 2012, Antifragile: Things that Gain from Disorder, New York: Random House, pp. 339, 346, 347, and 358; see likewise Silver, Nate, 2012, The signal and the noise: why so many predictions fail—but some don’t, New York: the Penguin Press, p. 183. The Economist article is “Unreliable research: Trouble at the lab, Scientists like to think of science as self-correcting. To an alarming degree, it is not,” October 19, 2013.
- 6.
- 7.
See Freedman, David and Diana B. Pettitti, 2002, “Salt, Blood Pressure, and Public Policy,” International Journal of Epidemiology, vol. 31 (2002), pp. 312–320.
- 8.
See Fisher, Ronald Aylmer, 1944, Statistical methods for research workers, London: Oliver and Boyd Ltd., ninth edition, p. 11.
- 9.
See Fisher, Ronald Aylmer, pp. 10, 41, 92, and 93.
- 10.
See Fisher, Ronald Aylmer, pp. 9, 19, and 21.
- 11.
See Silver, Nate, 2012, op. cit., pp. 251ff.; see also Wikipedia, “Kolmogorov-Smirnov test,” accessed February 27, 2013.
- 12.
See Freedman, David, 1995, “Some issues in the foundation of statistics,” Foundations of Science, vol. 1, pp. 19–83, reprinted in Some Issues in the Foundation of Statistics, Kluwer, Dordrecht (1997), Bas C. van Frassen, ed., p. 11.
- 13.
See Kaye, David H. and David A. Freedman, eds., 2011, “Reference Guide on Statistics,” pp. 211–301, in Reference Manual on Scientific Evidence, Washington, D. C., National Research Council, Committee on the Third Edition of the Reference Manual on Scientific Evidence & Federal Judicial Center, pp. 115, 124, and 125. This article contests the view in The Economist, 2013, op. cit., to the effect that a 5 % significance level has some probability meaning, such as that there is a 5 % chance that the hypothesis is incorrect.
- 14.
S. McGrayne, op. cit., cites various Bayesians on these issues on, for instance, pp. 53 and 132.
- 15.
Hill, Theodore P., and David E. R. Sitton, 2004, “Constructing Random Probability Distributions,” Abstract and Applied Analysis, 453–468, Chuong, Nurenberg, and Tutschek, ed., World Scientific Press. In order to construct various alternative distributions, merely normalize the results of Y= a0 + a1*x + a2*x**2 + … anx**n for any n so that the sum is 1.0 for x in U(0,1). One may use ai ≥ 0 to simplify the process. As higher-order equations are used, chances are increased that extreme value distributions can be produced, with their attendant “stability” issues.
- 16.
See Kaye, David H. and David A. Freedman, 2011, Ibid., p. 120.
- 17.
See Menke, W., 1989, Geophysical Data Analysis: Discrete Inverse Theory, San Diego: Academic Press, pp. 39–49 and 89, 90.
- 18.
See Silver, Nate, 2012, op. cit., pp. 163 and 166–167.
- 19.
See Silver, Nate, 2012, Ibid., pp. 251–261; see also Hanson, Norwood Russell, 1965, Patterns of Discovery: An Inquiry into the Conceptual Foundations of Science, Cambridge: Cambridge University Press.
- 20.
- 21.
See Menke, W., 1989, pp. 133 and 141.
- 22.
See Nolan, John P., 2009, Stable Distributions: Models for Heavy Tailed Data, accessed on the Internet February 27, 2013, p. 15, and Wikipedia, “Stable Distribution,” accessed February 27, 2013.
- 23.
See Silver, Nate, 2012, Ibid., pp. 251–261; see also Hanson, Norwood Russell, 1965, Patterns of Discovery: An Inquiry into the Conceptual Foundations of Science, Cambridge: Cambridge University Press.
References
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Taylor, C.E. (2015). Battling Inductivist vs. Deductivist Theories (1900s to Present). In: Robust Simulation for Mega-Risks. Springer, Cham. https://doi.org/10.1007/978-3-319-19413-4_6
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