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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.

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    $\begingroup$ You coukd get a null distribution by using a permutation test approach ... $\endgroup$ – kjetil b halvorsen♦ Nov 16 '16 at 16:22
  • $\begingroup$ @kjetilbhalvorsen, any article I can refer to? $\endgroup$ – John Nov 16 '16 at 16:27
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    $\begingroup$ Perhaps this Q&A stats.stackexchange.com/questions/61026/… $\endgroup$ – mdewey Nov 16 '16 at 16:32
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    $\begingroup$ 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. $\endgroup$ – kjetil b halvorsen♦ Nov 16 '16 at 16:52
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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.

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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)
y <- rnorm(100)
cor.test(x, y)
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