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I'm looking for a function with a set of properties and I don't know what to use. Anyone have any clues? It needs to output a probability, range [0,1) (note: strictly not equal to 1 at the upper limit), domain [0,infinity). It needs to satisfy the INADA conditions: f'(x)>0, f''(x)<0, f'(x)->infinity as x->0, f'(x)->0 as x->infinity.

The usual quartile functions (probit, logit) are sigmoid (S-shaped) and so don't satisfy the INADA conditions as they are convex and concave in different domains. Continuously differentiable everywhere the function is defined would be amazing. I feel like I'm missing something obvious, but any help appreciated :)

All the best, Yuan

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  • $\begingroup$ Note that "Inada" is a person's name, not an acronym -- so there's no need to shout it. $\endgroup$ – Henning Makholm Apr 5 '13 at 14:43
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How about $x\mapsto \sqrt{\frac{x}{1+x}}$?

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  • $\begingroup$ Alternatively, for a more gradual approach to 1, there's $\frac{\sqrt{x}}{1+\sqrt{x}}$. $\endgroup$ – Glen O Apr 5 '13 at 14:42

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