Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 30172
Approximations to the Distribution of the Sample Correlation Coefficient

Authors: John N. Haddad, Serge B. Provost

Abstract:

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.

Keywords: Sample correlation coefficient, density approximation, confidence intervals.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1334934

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1676

References:


[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.
[4] H. Hotelling, New light on the correlation coefficient and its transforms. Journal of Royal Statistical Society, Ser. B., 1953, 15: 193-232.
[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.
[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.