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I calculated the bias and variance of sample mean $\hat{\mu}$ and sample variance $\hat{\sigma}^2$ but I could not calculate the bias and variance for sample correlation $\hat{\rho}$. How can I do the calculation? Can you give me some references, textbooks, articles about it?

Edit: by sample correlation I mean $$ \hat{\rho}_{xy} = \frac{\sum_{i} x_i y_i}{\sqrt{\sum_{i}x_i^2 \sum_{i}y_i^2}} $$

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  • $\begingroup$ Can you explain what you mean by bias, and/or how you calculated it? $\endgroup$
    – Jeremy Miles
    Commented May 8, 2014 at 22:55
  • $\begingroup$ By bias I mean the difference between the estimator's expected value and the true value of the parameter being estimated $\endgroup$
    – neticin
    Commented May 9, 2014 at 9:15

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A basic into text should give the variance (is this what you mean?): $\sigma^{2}_{\rho}=\frac{1-\rho^{2}}{n-2} \approx \frac{1-\hat{\rho}^{2}}{n-2}$

Fisher addresses the bias: Fisher, R.A. (1915). Frequency distribution of the values of the correlation coefficient in samples from an indefinitely large population. Biometrika, 10, 507-521.

For a fuller treatment see: Zumbo, B. D., Williams, R. H., and Zimmerman, D. W. (2003). Bias in estimation and hypothesis testing of correlation. Psicológica: Revista de metodología y psicología experimental, 24(1):133– 158.

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  • $\begingroup$ In fact, I mean sample correlation between two variables x, y $\endgroup$
    – neticin
    Commented May 9, 2014 at 9:06
  • $\begingroup$ @Alexis good answer. Is there any bias in rho ρ ? If it is there, please explain to me. $\endgroup$
    – user10619
    Commented May 9, 2014 at 14:24
  • $\begingroup$ @neticin look at the last term of my equation on the first line, that's the sample correlation. $\endgroup$
    – Alexis
    Commented May 9, 2014 at 14:47
  • $\begingroup$ @Alexis can you point me a textbook where I can find this one because I didn't understand why the first equality is true $\endgroup$
    – neticin
    Commented May 9, 2014 at 15:41
  • $\begingroup$ Sure: Glantz, S. A. (2005). primer of biostatistics. McGraw-Hill Medical, New York, NY, 6th edition. $\endgroup$
    – Alexis
    Commented May 9, 2014 at 16:06

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