I have two sets of random data $X=\{x_1,...,x_N\}$ and $Y\{y_1,...,y_N\}$ both of length $N$. The sets are autocorrelated such that the correlation between $x_i$ and $x_j$ depends only on $|i-j|$. From both of these I can find the sample mean, $$ \bar{X} = \frac{1}{N}\sum_{i=1}^N x_i $$ and similarly for $Y$. I believe I can find the variance of each mean as follows, $$ \text{var}(\bar{X}) = \frac{S^2}{N}\frac{N-1}{\frac{N}{\gamma_2}-1} $$ where $S^2 = \frac{1}{N}\sum_{i=1}^N(x_i-\bar{X})^2$ and $\gamma_2 = 1+2\sum_{j=1}^{N-1}(1-j/N)\rho_j$, with $\rho_j$ being the autocorrelation function at a lag $j$. My question is this. What is $\text{cov}(\bar{X},\bar{Y})$ if the data sets are also correlated? The correlation between the sets should be the same for all pairs i.e. $\text{cov}(x_i,y_i) = \text{cov}(x_j,y_j)$. Will it be some generalisation of $\text{var}(\bar{X})$ or can I calculate it from the variances of the means and the correlation alone?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.