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Estimating the Correlation in Bivariate Normal Data With Known Variances and Small Sample Sizes

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  • Bailey K. Fosdick
  • Adrian E. Raftery

Abstract

We consider the problem of estimating the correlation in bivariate normal data when the means and variances are assumed known, with emphasis on the small sample case. We consider eight different estimators, several of them considered here for the first time in the literature. In a simulation study, we found that Bayesian estimators using the uniform and arc-sine priors outperformed several empirical and exact or approximate maximum likelihood estimators in small samples. The arc-sine prior did better for large values of the correlation. For testing whether the correlation is zero, we found that Bayesian hypothesis tests outperformed significance tests based on the empirical and exact or approximate maximum likelihood estimators considered in small samples, but that all tests performed similarly for sample size 50. These results lead us to suggest using the posterior mean with the arc-sine prior to estimate the correlation in small samples when the variances are assumed known.

Suggested Citation

  • Bailey K. Fosdick & Adrian E. Raftery, 2012. "Estimating the Correlation in Bivariate Normal Data With Known Variances and Small Sample Sizes," The American Statistician, Taylor & Francis Journals, vol. 66(1), pages 34-41, February.
    • Handle: RePEc:taf:amstat:v:66:y:2012:i:1:p:34-41
    DOI: 10.1080/00031305.2012.676329
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    1. James B. McDonald, 2008. "Some Generalized Functions for the Size Distribution of Income," Economic Studies in Inequality, Social Exclusion, and Well-Being, in: Duangkamon Chotikapanich (ed.), Modeling Income Distributions and Lorenz Curves, chapter 3, pages 37-55, Springer.
    2. John Carroll, 1961. "The nature of the data, or how to choose a correlation coefficient," Psychometrika, Springer;The Psychometric Society, vol. 26(4), pages 347-372, December.
    3. John C. Liechty, 2004. "Bayesian correlation estimation," Biometrika, Biometrika Trust, vol. 91(1), pages 1-14, March.
    4. Press, S. James & Zellner, Arnold, 1978. "Posterior distribution for the multiple correlation coefficient with fixed regressors," Journal of Econometrics, Elsevier, vol. 8(3), pages 307-321, December.
    5. Leontine Alkema & Adrian Raftery & Patrick Gerland & Samuel Clark & François Pelletier & Thomas Buettner & Gerhard Heilig, 2011. "Probabilistic Projections of the Total Fertility Rate for All Countries," Demography, Springer;Population Association of America (PAA), vol. 48(3), pages 815-839, August.
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    1. Bailey Fosdick & Adrian E. Raftery, 2014. "Regional probabilistic fertility forecasting by modeling between-country correlations," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 30(35), pages 1011-1034.
    2. Fosdick, Bailey K. & Perlman, Michael D., 2013. "Covariate and Newton–Raphson adjustments for a normal correlation coefficient when the variances are known," Statistics & Probability Letters, Elsevier, vol. 83(12), pages 2627-2633.
    3. Jarjour, Riad & Chan, Kung-Sik, 2020. "Dynamic conditional angular correlation," Journal of Econometrics, Elsevier, vol. 216(1), pages 137-150.
    4. Alan D. Hutson & Gregory E. Wilding & Terry L. Mashtare & Albert Vexler, 2015. "Measures of biomarker dependence using a copula-based multivariate epsilon-skew-normal family of distributions," Journal of Applied Statistics, Taylor & Francis Journals, vol. 42(12), pages 2734-2753, December.
    5. Veronese, Piero & Melilli, Eugenio, 2018. "Some asymptotic results for fiducial and confidence distributions," Statistics & Probability Letters, Elsevier, vol. 134(C), pages 98-105.

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