How do you compute the sample variance between the difference in means of two samples?

Question

I am given the means for four samples $x_1,...x_4$ and a sample covariance matrix S that’s size $4\times 4$.

How do I compute the sample covariance for $x_1-x_2$? I assumed it would just be $s_{11}-s_{22}$, but I am not getting the correct value.

Answer 1

If you are given a $S \in \Bbb{R^{4 \times 4}}$ covariance matrix, then each element will provide you with the variance between $x_i$ and $x_j$. $$Cov(x_i,x_j) =s_{ij}$$ where $S = [s_{ij}]_{i,j=1}^4$.

Another important fact to note is that $Var(x_i-x_j) = Var(x_i) + Var(x_j) - 2\ Cov(x_i,x_j)$. (See more basic variance properties here.)

And lastly, $Var(x_i) = Cov(x_i,x_i) = s_{ii}$.

Now putting these all together:

$$Var(x_i-x_j) = Var(x_i) + Var(x_j) - 2\ Cov(x_i,x_j) = s_{ii} + s_{jj} - 2\ s_{ij}$$

Or, since the covariance matrix is symmetric:

$$Var(x_i-x_j) = Var(x_i) + Var(x_j) - 2\ Cov(x_i,x_j) = s_{ii} + s_{jj} - s_{ij} - s_{ji}.$$

And specifically,

$$Var(x_1-x_2) = Var(x_1) + Var(x_2) - 2\ Cov(x_1,x_2) = s_{11} + s_{22} - 2\ s_{12}.$$