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I'm looking for a 2 (or more)-parameter family of functions $F$ with the following properties:

  1. For each $f \in F$, $f(0)=0$, $f(1)=1$, and $f$ is (weakly) increasing.

  2. $F$ is closed under products.

  3. $F$ is closed under the transformation $f(x) \rightarrow 1-f(1-x)$.

Equivalently, $F$ is a family of cumulative distribution functions for a family of random variables which are closed under maximum and the transformation $X \rightarrow 1-X$.

Note: One obvious solution is the set of polynomials satisfying (1). However, these become very large under multiplication. I would like each function to have a compact $n$ parameter representation for some constant $n$, and for the transformations in (2) and (3) to have a simple form in the parameter space.

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  • $\begingroup$ If F contains an absolutely continuous member g, I don't think you can avoid the polynomials. Your problem is then essentially unchanged under the transformation X → g(X), and thus the transformed space contains the uniform CDF; h(x) = x on (0,1). This pulls a lot of polynomials into the transformed space, and their inverse images into the original. $\endgroup$
    – guest
    Aug 31, 2014 at 9:14

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