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Consider I have realisations of two random variables $X$ and $Y$ and I estimate their correlation thanks to the classic formula :

$$\rho=\frac{\sum_{i=1}^{n}{x_iy_i}-\sum_{i=1}^{n}x_i\sum_{i=1}^{n}y_i}{\sqrt{\sum_{i=1}^{n}(x_i-\bar{x})^2\sum_{i=1}^{n}(y_i-\bar{y})^2}}$$

1- What is the variance of the estimator and how to calculate it?

2- Hypothesis tests use the t-statistic:

$$t=\frac{r\sqrt{n-2}}{1-r^2}$$

Why is that?

Any references or pointers to pdfs, courses, or textbooks with exhaustive treatments are welcome.

Thanks for help

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  • $\begingroup$ Please share your thoughts so far :) $\endgroup$ – Shaun Jul 8 '14 at 8:07
  • $\begingroup$ OP - your definition of the sample correlation coefficient appears incorrect. $\endgroup$ – wolfies Jul 8 '14 at 15:01
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The variance of the sample correlation is not an easy question; nor is an easy general answer available. If you refer to:

  • Stuart and Ord (1994), Kendall's Advanced Theory of Statistics, volume 1 - Distribution Theory, sixth edition, Edward Arnold

... an approximation is provide at eqn (10.17) (based on what is essentially the delta method) to the variance of a ratio $\frac{S}{U}$ (provided $S$ and $U$ are positive):

$$Var\big(\frac{S}{U}\big) \approx \big(\frac{E[S]}{E[U]}\big)^2 \big( \frac{Var(S)}{(E[S])^2} + \frac{Var(U)}{(E[U])^2} - \frac{2 Cov(S,U)}{E[S] E[U]} \big) $$

In your instance,

$$S = m_{11} \quad \quad \text{and } \quad \quad U = \sqrt{m_{20} m_{02}}$$

where $m_{ab}$ denotes the sample central product moments.

They then apply this approximation in their Example 10.6 to find an approximation to the variance of the sample correlation ... which is what you seek. The $\sqrt{}$ in $U$ poses further difficulties ... they appear to deal with the $\sqrt{m_{20} m_{02}}$ in the denominator by a further approximation (again via the delta method). In any event, an approximate solution is posited on these pages to a non-trivial problem as:

$$Var(r) \approx \frac{\rho^2}{n} \big( \frac{\mu_{22}}{\mu_{11}^2} + \frac14 \big(\frac{\mu_{40}}{\mu_{20}^2} + \frac{\mu_{04}}{\mu_{02}^2} + \frac{2 \mu_{22}}{\mu_{20} \mu_{02}}\big) - \big( \frac{\mu_{31}}{\mu_{11}\mu_{20}} + \frac{\mu_{13}}{\mu_{11}\mu_{02}} \big) \big) $$

where $r$ denotes the sample correlation coefficient, $\rho$ denotes the population correlation coefficient, and $\mu_{ab}$ denotes product central moments of the population.

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