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Paul
Posts:
208
Registered:
2/23/10
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Jensen's inequality & convexity of "max" function
Posted:
Oct 3, 2012 10:24 AM
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In the context of the financial value ascribed to the ability to hold
off on deciding whether to make an investment, Jensen's inequality has
been used to describe the fact that the average of the maximum returns
from several possible future outcomes is greater than the maximum of
the average returns for those same set of possible future outcomes.
This is because (apparently) the max function is convex. However,
I've only seen Jensen's inequality described in terms of a function
that takes one input argument and returns one output argument i.e. a
transformation of the outcomes values of a random variable. The max
function is not like that. So it's not possible to plot the max
function as output value versus input value. For this reason, the
justification for calling the max function convex eludes me. In what
sense is the max function convex?
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