Re-parameterization of an asymmetric s-shaped function Is anyone aware of a re-parameterization of any asymmetric s-shaped function (like, but not necessarily the 5 parameter logistic curve), where one of the parameters is the first inflection point of the first derivative (i.e. the maximum of the second derivative).
I mean point 1 in the upper figure:

(The picture shows the point mentioned above for the of case of a symmetric logistic function.)
So far, I had a look at various references (among others Ratkowsky, D. A. (1990). Handbook of Nonlinear Regression Models. New York, Dekker). However, I have only found parameterizations, where one of the parameters is the inflection point of the function (point 2 in the upper figure). 
Unfortunately, calculating this point after the estimation is not a possible solution in my case as this parameter should be integrated in another equation that is estimated simultaneously.
 A: Pace Procrastinator, this sort of thing can be done.
Consider the five-parameter logistic model.  It has many parameterizations, but it's simple and natural to reduce them to something like
$$y = \nu +\tau  \left(1+e^{\frac{x-\mu }{\sigma }}\right)^{-\rho }$$
with $\rho \gt 0$.  We can interpret $\mu$ and $\nu$ as $x$ and $y$ locations and $\sigma$ and $\tau$ as $x$ and $y$ scales; $\rho$ is the shape or asymmetry parameter.
Let $x_{+}$ be the location of one extremum of the second derivative and $x_{-}$ be the location of the other extremum.  Then, solving for the zeros of the third derivative, I find them at
$$x_{\pm} = \mu + \sigma  \log \left(\frac{(3 \rho +1) \pm\sqrt{(\rho +1) (5 \rho +1)
  }}{2 \rho ^2}\right).$$
Whence, setting 
$$\kappa = \exp \frac{x_{+}-x_{-}}{\sigma}$$
we find
$$\rho = \frac{3 \kappa \pm\sqrt{\kappa ^3+2 \kappa ^2+\kappa }}{\kappa ^2-7 \kappa +1},$$
taking the positive sign when $\kappa$ exceeds the larger root of $1-7x+x^2=0$ (about $6.8541$) and the negative sign when $\kappa$ is less than the smaller root (about $0.145898$)--other values of $\kappa$ will not give a sigmoidal curve--and 
$$\mu = \frac{x_{+} + x_{-}}{2} + \sigma \log(\rho).$$
This allows a parameterization in terms of $(\sigma, \nu, \tau, x_{-}, x_{+})$ (with some significant restrictions on $\sigma$ needed to make the results valid).
Here is a plot of $y$ (in blue) and its third derivative (in red) based on these formulas with the parameters set to $(-1/4, 1/2, 1, 1, 0)$:

Indeed, the ascending zero of the third derivative occurs at $1$ and the descending zero at $0$, as specified.

This parameterization is not necessarily so messy.  If your data are within a narrow range, for instance, the asymptotic values might not matter and you could just use a suitable polynomial.  Without knowing more about the particular problem, it's hard to provide a specific recommendation.
