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A History of Parametric Statistical Inference from Bernoulli to Fisher, 1713–1935Credibility and Confidence Intervals by Laplace and Gauss

A History of Parametric Statistical Inference from Bernoulli to Fisher, 1713–1935: Credibility... [It follows from Laplace’s 1774 and 1785 papers that the large-sample inverse probability limits for θ are given by the relation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ P\left( {h - u\sqrt {h\left( {1 - h/n} \right)} < \theta < h + u\sqrt {h\left( {1 - h} \right)} /n|h} \right) \cong \Phi \left( u \right) - \Phi \left( { - u} \right), $$\end{document} for u > 0. In 1812 ([159], II, §16) he uses the normal approximation to the binomial to find large-sample direct probability limits for the relative frequency as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ P\left( {\theta - u\sqrt {\theta \left( {1 - \theta } \right)/n} < h < \theta + u\sqrt {\theta \left( {1 - \theta /n} \right)} |\theta } \right) \cong \Phi \left( u \right) - \Phi \left( { - u} \right). $$\end{document} Noting that θ = h + O(n−1/2) so that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sqrt {\theta \left( {1 - \theta \,} \right)/n} = \sqrt {h\left( {1 - h/n} \right)} + O\left( {n^{ - 1} } \right) $$\end{document} and neglecting terms of the order of n−1 as in the two formulas above he solves the inequality (8.2) with respect to θ and obtains for u > 0 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ P\left( {h - u\sqrt {h\left( {1 - h/n} \right)} < \theta < h + u\sqrt {h\left( {1 - h} \right)/n} |\theta } \right) \cong \Phi \left( u \right) - \Phi \left( { - u.} \right) $$\end{document}] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

A History of Parametric Statistical Inference from Bernoulli to Fisher, 1713–1935Credibility and Confidence Intervals by Laplace and Gauss

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Publisher
Springer New York
Copyright
© Springer Science+Business Media, LLC 2007
ISBN
978-0-387-46408-4
Pages
63 –66
DOI
10.1007/978-0-387-46409-1_8
Publisher site
See Chapter on Publisher Site

Abstract

[It follows from Laplace’s 1774 and 1785 papers that the large-sample inverse probability limits for θ are given by the relation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ P\left( {h - u\sqrt {h\left( {1 - h/n} \right)} < \theta < h + u\sqrt {h\left( {1 - h} \right)} /n|h} \right) \cong \Phi \left( u \right) - \Phi \left( { - u} \right), $$\end{document} for u > 0. In 1812 ([159], II, §16) he uses the normal approximation to the binomial to find large-sample direct probability limits for the relative frequency as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ P\left( {\theta - u\sqrt {\theta \left( {1 - \theta } \right)/n} < h < \theta + u\sqrt {\theta \left( {1 - \theta /n} \right)} |\theta } \right) \cong \Phi \left( u \right) - \Phi \left( { - u} \right). $$\end{document} Noting that θ = h + O(n−1/2) so that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sqrt {\theta \left( {1 - \theta \,} \right)/n} = \sqrt {h\left( {1 - h/n} \right)} + O\left( {n^{ - 1} } \right) $$\end{document} and neglecting terms of the order of n−1 as in the two formulas above he solves the inequality (8.2) with respect to θ and obtains for u > 0 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ P\left( {h - u\sqrt {h\left( {1 - h/n} \right)} < \theta < h + u\sqrt {h\left( {1 - h} \right)/n} |\theta } \right) \cong \Phi \left( u \right) - \Phi \left( { - u.} \right) $$\end{document}]

Published: Jan 1, 2007

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