Aggregating risk scores

I have a model which calculates the risk of default on several essentially independent metrics. These are default probabilities between 0 to 1. 1 implying a high risk of default.

Now I wish to combine these scores into a single score. I naively started using an average of all scores but that doesn't give me the desired behavior when one score is very high and the others are low.

i.e. Say the component scores are 0.1,0.1, and 0.999. Essentially I want the aggregate score in this case to be close to 0.999 as well.

What might be a clean functional form to get this sort of behavior? i.e. When individual scores are low an average score is OK but when any one score shoots above a threshold value that score ought to dominate the aggregate.

PS. If someone has an R idiom for this then I am even happier.

• You appear to be describing the maximum. However, your question does not supply enough information to determine and objectively support any particular answer. You are essentially asking your readers to guess the results of a modeling exercise in which you relate these metrics to actual default experiences. If you don't have such data, then we're all just guessing. Do you have relevant data? What are their nature?
– whuber
Jul 30, 2015 at 13:04
• @whuber I'll edit my question. No, it is not the maximum because when the individual scores are below a thresh-hold the averaging is the right behavior. It is only above the threshold that I wanted them any high value to swamp the rest. Maybe what I am needing is a functional form that transitions from the average to the maximum smoothly. Jul 30, 2015 at 13:17
• @whuber No I don't want the readers the guess the results of the models. I am only hoping someone can suggest a functional form with these properties. Jul 30, 2015 at 13:18
• They have no basis to suggest a functional form. Your question is asking readers to make pure guesses. (There are infinitely many possible such functions.) You need to supply sufficient information to allow people either to determine answers or to be able to explain how you could use that information to find an answer yourself.
– whuber
Jul 30, 2015 at 13:20
• @whuber OK. I will contemplate what else I can provide. Jul 30, 2015 at 15:21

$(1 - score)^n = \prod_{i=1}^n{(1 - x_i)}$