Summary
The problem of the inferential analysis of the linear correlation coefficient of normal bivariate populations is tackled, both from the likelihood and Bayesian viewpoints. In particular it is shown how, using pseudo-likelihood (marginal likelihood function and profile likelihood), hypotheses such asH 0:ϱ=ϱ0 andH 0:ϱx=ϱy can be verified without prohibitive computation effort. The results of marginal and profile likelihood are compared and it is shown that these two methods are virtually equivalent even for small sample sizes. Furthermore, in suitable conditions, the posterior distribution of the coefficient ϱ can be readily obtained, using the exact form or different approximate formulations of the marginal or profile likelihood. Lastly some possible prior distributions of ϱ are illustrated and some explanatory examples are presented.
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References
Basu D. (1977), On the elimination of nuisance parameters,JASA, 72, 355–366.
Bayarri M. J. (1981), Inferencia Bayesiana sobre el coeficiente de correlation de una poblacion normal bivariante,Trabajos de Estadistica y de Investigacion Operativa, 32, 18–31.
Bernardo J. M. (1979), Reference posterior distributions for Bayesian inference,J. R. Statist. Soc. A, 41, 113–28.
Bertolino F., Piccinato L., Racugno W. (1990), A marginal likelihood approach to analysis of variance,The Statistician, 39, 415–24.
Box G. E. P., Tiao G. C. (1973),Bayesian Inference in Statistical Analysis., Reading, MA, Addison Wesley.
Cox D. R. (1975), Partial likelihood,Biometrika, 62, 269–276.
Cramér H. (1946),Mathematical Methods of Statistics, Princeton University Press.
David F. N. (1938),Tables of the Correlation Coefficient, Cambridge University Press, London.
Edwards A. W. F. (1976),Likelihood, Cambridge University Press, London.
Gokhale D. V., Press S. J. (1982), Assessment of a prior distribution for the correlation coefficient in a bivariate normal distribution,J. R. Statist. Soc. A, 145, 237–49.
Hotelling H. (1953), New light on the correlation coefficient and its transforms,J. R. Statist. Soc. B, 15, 193–224.
Jeffreys H (1961),Theory of Probability (3rd edn.), Oxford University Press, London.
Johnson N. L., Kotz S. (1970),Continuous Univariate Distributions-2, Wiley, N. Y.
Kalbfleisch J. D. (1986),Pseudo-likelihood, In Enc. of Statistical Science, S. Kotz, N. L. Johnson, C. B. Read, eds. vol. 7, Wiley, N. Y.
Lindley D. V. (1965),Introduction to Probability and Statistics from a Bayesian Viewpoint (part. 2), Cambridge University Press, Cambridge.
Smith A. F. M., Skene A. M., Shaw J. E. H., Naylor J. C. (1987), Progress with numerical and graphical methods for practical Bayesian statistics,The Statistician, 36, 75–82.
Stuart A., Ord J. K. (1987),Kendall's Advanced Theory of Statistics, (5th edn.), vol. 1, Distribution Theory, Griffin, London.
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Bertolino, F., Racugno, W. Analysis of the linear correlation coefficient using pseudo-likelihoods. J. It. Statist. Soc. 1, 33–50 (1992). https://doi.org/10.1007/BF02589048
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DOI: https://doi.org/10.1007/BF02589048