How can I test whether the mean of a sample is significantly larger than the population mean, when observations are not independent, but the exact correlation matrix is known?
- I have a variable $X$ for a finite population (a set of spatial units).
- The variable pair ($X_i$,$X_j$) is correlated when spatial units $i$ and $j$ are spatially close to each other.
- I can compute the population mean $\mu$, and variance $\sigma^2$, as well as the covariance matrix $\Sigma$ ($\Sigma$ is known for the entire population).
- I draw a sample from this population, and I would like to know whether the sample mean $\overline{X}$ is significantly larger than the population mean $\mu$.
- If variables $X_i$ were i.i.d., I could simply perform a Z-test.
- In my situation, the variables are not i.i.d (they are spatially correlated).
- I wonder if there is another test that I could use, given that I know the covariance matrix $\Sigma$, ($\sigma_{ij}^2$ is the covariance of $X_i$ and $X_j$).
Some more detail on the data:
- The variable $X$ is a spatial moving average of another variable $V$.
- $V$ is i.i.d, bounded between 0 and 1.
- The moving average can be expressed as follows: $X_i = \sum_j w_{ij} V_j$. An equivalent in time series would be $X_t= \sum_h w_{t+h}(t,h)\cdot V_{t+h}$.
- The coefficient $w_{ij}$ decreases as $1/d_{ij}^2$, with $d_{ij}$ the distance between spatial unit $i$ and $j$, and is set to 0 after a certain distance threshold.
- $X$ is not stationary: the covariance $Cov(X_i,X_{i+h})$ depends on $i$.
- When I sample the data, I take a set of contiguous spatial units - a region -, and I would like to state whether that region's average is significantly different.