# test zero correlation coefficient

Is there any commonly used method to test the zero correlation between $X$ and $Y$ using the sample correlation coeffcient from two samples $\{x_k\}_{k=1}^n$ and $\{y_k\}_{k=1}^n$?

In another word, if $\boldsymbol{x}=(x_1,\dots,x_n)^\top$ and $\boldsymbol{y}=(y_1,\dots,y_n)^\top$ are two standardized vectors, i.e., $\sum_{i=1}^nx_i=0$ and $\sqrt{\sum_i x_i^2}=1$, is there any method to numerically judge $\boldsymbol{x}\perp \boldsymbol{y}$?

Actually $\boldsymbol{x}$ and $\boldsymbol{y}$ are two observed noisy data vectors, so I do not think it is appropriate to use $\boldsymbol{x}^\top \boldsymbol{y}=0$ as the criterion.

• You coukd get a null distribution by using a permutation test approach ... – kjetil b halvorsen Nov 16 '16 at 16:22
• @kjetilbhalvorsen, any article I can refer to? – John Nov 16 '16 at 16:27
• Perhaps this Q&A stats.stackexchange.com/questions/61026/… – mdewey Nov 16 '16 at 16:32
• In Efron's book about the bootstrap, bootsrapping the corelation coefficient is the first example. For permutation methods: projecteuclid.org/euclid.ss/1113832732 Else, for a more useful answer, you shoukd tell us the context of your question. – kjetil b halvorsen Nov 16 '16 at 16:52

Assuming your observations follow a bivariate normal distribution, there's a test based on t-statistic for the Pearson correlation coefficient which gives the test statistic $$t = r \sqrt{\frac{n-2}{1-r^2}}.$$ This has a t-distribution with $n-2$ degrees of freedom.
If your x and y are reasonably normal and $n$ is big enough, you can use the cor.test function in R, like this:
x <- runif(100)