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Approximations To The Distribution Of The Sample Correlation Coefficient
NSERC
Haddad, John N.; Provost, Serge B.;
Haddad, John N.; Provost, Serge B.;
Approximations To The Distribution Of The Sample Correlation Coefficient
{"references": ["A. M. Mathai, The concept of correlation and misinterpretations. International\nJournal of Mathematical and Statistical Sciences, 1998, 7: 157-167.", "R. A. Fisher, Distribution of the values of the correlation coefficient in\nsamples from an indefinitely large population. Biometrika, 1915, 10: 507-\n521.", "A. Winterbottom, A note on the derivation of Fisher-s transformation of\nthe correlation coefficient. The American Statistician, 1979, 33: 142-143.", "H. Hotelling, New light on the correlation coefficient and its transforms.\nJournal of Royal Statistical Society, Ser. B., 1953, 15: 193-232.", "A. K. Gayen, The frequency distribution of the product-moment correlation\ncoefficient in random samples of any size drawn from non-normal\nuniverses. Biometrika, 1951, 38: 219-247.", "D. L. Hawkins, Using U statistics to derive the asymptotic distribution\nof Fisher-s Z statistic. The American Statistician, 1989, 43: 235-237.", "S. Konishi, An approximation to the distribution of the sample correlation\ncoefficient. Biometrika, 1978, 65: 654-656.", "H.-T. Ha and S. B. Provost, A viable alternative to resorting to statistical\ntables. Communications in Statistics-Simulation and Computation, 2007,\n36: 1135-1151."]}
Given a bivariate normal sample of correlated variables, (Xi, Yi), i = 1, . . . , n, an alternative estimator of Pearson's correlation coefficient is obtained in terms of the ranges, |Xi − Yi|. An approximate confidence interval for ρX,Y is then derived, and a simulation study reveals that the resulting coverage probabilities are in close agreement with the set confidence levels. As well, a new approximant is provided for the density function of R, the sample correlation coefficient. A mixture involving the proposed approximate density of R, denoted by hR(r), and a density function determined from a known approximation due to R. A. Fisher is shown to accurately approximate the distribution of R. Finally, nearly exact density approximants are obtained on adjusting hR(r) by a 7th degree polynomial.
density approximation, confidence intervals., Sample correlation coefficient
density approximation, confidence intervals., Sample correlation coefficient
[1] A. M. Mathai, The concept of correlation and misinterpretations. International Journal of Mathematical and Statistical Sciences, 1998, 7: 157-167.
[2] R. A. Fisher, Distribution of the values of the correlation coefficient in samples from an indefinitely large population. Biometrika, 1915, 10: 507- 521.
[3] A. Winterbottom, A note on the derivation of Fisher's transformation of the correlation coefficient. The American Statistician, 1979, 33: 142-143. [OpenAIRE]
[4] H. Hotelling, New light on the correlation coefficient and its transforms. Journal of Royal Statistical Society, Ser. B., 1953, 15: 193-232. [OpenAIRE]
[5] A. K. Gayen, The frequency distribution of the product-moment correlation coefficient in random samples of any size drawn from non-normal universes. Biometrika, 1951, 38: 219-247. [OpenAIRE]
[6] D. L. Hawkins, Using U statistics to derive the asymptotic distribution of Fisher's Z statistic. The American Statistician, 1989, 43: 235-237.
[7] S. Konishi, An approximation to the distribution of the sample correlation coefficient. Biometrika, 1978, 65: 654-656.
[8] H.-T. Ha and S. B. Provost, A viable alternative to resorting to statistical tables. Communications in Statistics-Simulation and Computation, 2007, 36: 1135-1151.
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citations
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This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
Citations provided by BIP!popularityPopularity provided by BIP!
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
Popularity provided by BIP!influenceInfluence provided by BIP!
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
Influence provided by BIP!impulseImpulse provided by BIP!
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
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48
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Funded by
NSERC
Project
- Natural Sciences and Engineering Research Council of Canada (NSERC)
{"references": ["A. M. Mathai, The concept of correlation and misinterpretations. International\nJournal of Mathematical and Statistical Sciences, 1998, 7: 157-167.", "R. A. Fisher, Distribution of the values of the correlation coefficient in\nsamples from an indefinitely large population. Biometrika, 1915, 10: 507-\n521.", "A. Winterbottom, A note on the derivation of Fisher-s transformation of\nthe correlation coefficient. The American Statistician, 1979, 33: 142-143.", "H. Hotelling, New light on the correlation coefficient and its transforms.\nJournal of Royal Statistical Society, Ser. B., 1953, 15: 193-232.", "A. K. Gayen, The frequency distribution of the product-moment correlation\ncoefficient in random samples of any size drawn from non-normal\nuniverses. Biometrika, 1951, 38: 219-247.", "D. L. Hawkins, Using U statistics to derive the asymptotic distribution\nof Fisher-s Z statistic. The American Statistician, 1989, 43: 235-237.", "S. Konishi, An approximation to the distribution of the sample correlation\ncoefficient. Biometrika, 1978, 65: 654-656.", "H.-T. Ha and S. B. Provost, A viable alternative to resorting to statistical\ntables. Communications in Statistics-Simulation and Computation, 2007,\n36: 1135-1151."]}
Given a bivariate normal sample of correlated variables, (Xi, Yi), i = 1, . . . , n, an alternative estimator of Pearson's correlation coefficient is obtained in terms of the ranges, |Xi − Yi|. An approximate confidence interval for ρX,Y is then derived, and a simulation study reveals that the resulting coverage probabilities are in close agreement with the set confidence levels. As well, a new approximant is provided for the density function of R, the sample correlation coefficient. A mixture involving the proposed approximate density of R, denoted by hR(r), and a density function determined from a known approximation due to R. A. Fisher is shown to accurately approximate the distribution of R. Finally, nearly exact density approximants are obtained on adjusting hR(r) by a 7th degree polynomial.