Jerry W. Lewis wrote:
It is not at all clear to me that the linear shift you propose (to deal with
negative values) is appropriate for dealing with this data.

It is appropriate, but it is not necessary. I had started with "1+g"
for my earlier response out of habit, because that __is__ necessary
when computing the log. But then I realized that SUM(w[i]*g[i]) =
SUM(w[i]*(1+g[i])) - 1.

But all this is academic with respect to the OP's problem because I
believe the OP is not dealing with a time series, but with a collection
of categories. As I explained in my earlier response, we can
demonstrate that in that case, SUM(w[i]*g[i]) is the correct solution
for the OP's problem, and it does matter that some g[i] are negative.