I'm NOT asking about the "usual" weighted arithmetic mean where each observation $x_i$ has an associated weight $w_i$, and the weighted arithmetic mean of the series is defined as:
$\bar{x}=\frac{\sum_i{w_ix_i}}{\sum_i{w_i}}$
Instead, what if there are 2 or more weights associated with each observation, i.e. instead of an Nx1 vector of weights for the Nx1 vector of observations, there is now an NxM matrix of weights.
I can transform each of the individual weight column vectors to be normalized by dividing by the sum of that vecotr of weights (so each element in a given column of the matrix is divided by the sum of that column). But then if I naively combined the weights as follows, I would essentially just be taking a a weighted arithmetic mean of the product of the weights, which isn't necessarily what I want:
$\bar{x}=\frac{\sum_i(\prod_j{{w_{ij})x_i}}}{\sum_i(\prod_j{{w_{ij})}}}$, e.g. for 2 weights per observation: $\bar{x}=\frac{\sum_i{{w_{i1}w_{i2}x_i}}}{\sum_i{{w_{i1}w_{i2}}}}$
Is there a standard way of dealing with a weighted arithmetic mean when each observation has 2 to N weights?