Abstract
In probability theory, we always assumed that we knew some probabilities, and we computed other probabilities or related quantities from those. On the other hand, in mathematical statistics, we use observed data to compute probabilities or related quantities or to make decisions or predictions.
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Notes
- 1.
The requirement that an estimator be unbiased can lead to absurd results. See M. Hardy, An illuminating counterexample, Am. Math. Monthly 110 (2003) 234–238.
- 2.
Some people use a slightly different terminology. They call the random interval \(\left (A,B\right )\) a confidence interval and \(\left (a,b\right )\) its observed value.
- 3.
Of course, a probability cannot be greater than 1, and so the upper limit of the interval should be 1 rather than \(\infty,\) but the normal approximation gives only a minuscule probability to the (1,\(\infty )\) interval, and we may therefore ignore this issue.
- 4.
However, some statisticians prefer to say “do not reject H 0” rather than “accept H 0” when the test gives just weak evidence in favor of H 0, and they imply or say that the test is inconclusive or that further testing is needed.
- 5.
In some books, the rejection region is defined to be the set in the n-dimensional space of sample data that corresponds to C.
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A. A. Lelis, W. A. Percy and B. C. Verstraete, The Age of Marriage in Ancient Rome (The Edwin Mellen Press, 2003)
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J. L. Hopper and E. Seeman, The Bone Density of Female Twins Discordant for Tobacco Use. NEJM, Feb. 14, 1994.
- 8.
A rough rule of thumb is that n should be large enough so that n p 0i ≥ 10 for each i, although some authors go as low as 5 instead of 10 and, for large k, even allow a few n p 0i to be close to 1.
- 9.
The result could be explained by non-independent distributions as well, but a computation of type 2 errors would be hopeless because of the various ways non-independence can occur.
- 10.
D.O. Clegg et al. Glucosamine, Chondroitin Sulfate, and the Two in Combination for Painful Knee Osteoarthritis. NEJM. Feb. 2006.
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Randomized controlled trial of the Lidcombe programme of early stuttering intervention. Mark Jones et al., BMJ Sep. 2005.
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Statistics for Analytical Chemistry, J.C. Miller and J. N. Miller.
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Acute treatment of moderate-to-severe depression with hypericum extract WS 5570 (St John’s wort): randomised controlled double blind non-inferiority trial versus paroxetine
A Szegedi, R Kohnen, A Dienel, M Kieser, BMJ, March 2005.
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R. A. Fisher, Statistical Methods for Research Workers. Oliver & Boyd, 1925. Also available at http://psychclassics.yorku.ca/Fisher/Methods/index.htm
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M Gulati & al. The Prognostic Value of a Nomogram for Exercise Capacity in Women. N Engl J Med 2005; 353:468-475
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Schay, G. (2016). The Elements of Mathematical Statistics. In: Introduction to Probability with Statistical Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-30620-9_8
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DOI: https://doi.org/10.1007/978-3-319-30620-9_8
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