Why do weights in a weighted average have to be positive?
I'm not sure there's any "law" against using negative weights. How would you interpret a negative weight?
In the various pages I found in a Google search, i.e., Wikipedia, etc .. in defining weights they all specified that the weights must be non-negative.
Suppose you are looking at the profit and loss statement of a foreign company over a period of time. You want to convert the foreign currency profit or loss to US dollars over, say a quarter. Over that time you can compute an arithmetic average of the FX rates. You could also weight the exchange rates with net income of the company. However, the company may have had months (or days, whatever you are using) with net income and other months with net loss. So the loss periods would be negative weights.
If you want to weight the contributions of items in some kind of compilation in terms of the magnitude of profit or loss, I see no reason why you couldn't do that. Essentially just summing over negative and positive numbers does exactly that. Quoting the Wiki needs to be done with care. It's context specific and not always reliable. Technically, calculating the mean or average can be an issue for mathematical types because you can get a sum of zero which you then must divide by the number of values less 1 to get the mean. However, meteorologists deal with negative temperature values all the time and are able to calculate means.
Weights are usually associated with probabilities, which is why I don't believe they can be negative. The number you are weighting, can of course be negative.
Weights are used in non probabilistic contexts such as averaging over ordered categorical data where the numbers in each category vary. It might be useful to give signed weights based on whether the category is seen as enhancing or detracting in some evaluative context.
Hmm, could you give a simple example of what you mean?
Family income strata weighted by size and "voting score" based on log likelihood of voting for a given party where 0 is neutral.
I guess if you weren't concerned with something physically meaningful coming out of your average you could assign weight whatever value you wanted.
Let me try and throw out a simple example, and maybe you can modify it to what you mean. Lets take a number like family debt, and average it over families with children. Now instead of weighting by number of families (or some sensible weight), we weight by (#ofChildren - 2.5), or whatever the average number of children is. So families with 2 children appear to have negative debt (or positive money), while families with 3 children have positive debt (and negative money).
I don't think that example makes meaningful sense, on account of the negative weights. When I think of weighting a sample, its statistical weights that are being added. But maybe that's just limited thought.
Now if you wanted to create a model describing family debt, and assign a positive or negative model parameter times the number of children, then that would of course work. But I wouldn't call a model parameter a weight.
Anyways sorry if I'm not getting it, or if this is all just semantics, but off hand I'd side with Wikipedia and the internets on this one.
An average is a "middle" value, but if any weights are negative it is possible for the result to be outside the range of the data - so it would be called a "linear combination" instead.
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