Is there a standard/frequently used/convenient equivalent to sigmoid function with two thresholds?
Background
When writing a likelihood of data for a binary classification problem one would often parametrize the probability to be in a class by a sigmoid function:
$$
P(y=1|x,a,b)=\frac{1}{1+e^{-ax-b}}, P(y=0|x,a,b)=\frac{1}{1+e^{ax+b}},
$$
which roughly corresponds to classification rule
$$
y=\begin{cases}
1\text{ if } ax+b>0\Leftrightarrow x > -\frac{b}{a},\\
0\text{ if } ax+b<0\Leftrightarrow x < -\frac{b}{a},
\end{cases}
$$
that is we compare a predictive variable to a single threshold $\mu=-\frac{b}{a}$
Objective
Now, I would like to have a smooth representation, corresponding to a classification with two thresholds:
$$
y=\begin{cases}
1\text{ if } \mu < x<\nu,\\
0\text{ if } x<\mu \text{ or } x>\nu
\end{cases}
$$
Options
Some of the possibilities that come to mind are:
product of sigmoids
$$
P(y=1|x)=\frac{1}{1+e^{a(x-\nu)}}\frac{1}{1+e^{-b(x-\mu)}}
$$
difference of sigmoids
$$
P(y=1|x)=\frac{1}{1+e^{-a(x-\nu)}}-\frac{1}{1+e^{b(x-\mu)}}
$$
sigmoid of a more complex argument
$$
P(y=1|x)=\frac{1}{1+e^{a(x-\nu)(x-\mu)}}
$$
I wonder, whether any of these or some other representation is commonly used for such classification tasks and what are the potential advantages or disadvantages.
