Logistic

Parameters	scale ($b$)
CDF	${\left\{ {1 + \exp ( - x/b)} \right\}^{ - 1}}$
Kernel	$frac{{\exp ( - x/b)}}{{b{{\left( {1 + \exp ( - x/b)} \right)}^2}}}$
Support	$\left[ -\infty,\infty \right) $
Mean	0. (A non-zero mean is a relatively easy change)
Variance	${\pi ^2}{b^2}/3$
Main Uses	Probability in logit model, transition probability mapping in Markov Switching models, transition function in LSTAR models.
Density Function	%LNLOGISTIC(x) is the log density with b=1 (most common case)
CDF Function	%LOGISTIC(x,b) is logistic CDF of x with scale b. You should use this rather than computing directly, as %LOGISTIC does the calculation in an "overflow-safe" fashion. (In the direct calculation, a large negative $x$ will cause ${\exp ( - x/b)}$ to overflow).

Parameters	scale ($b$)
CDF	\({\left\{ {1 + \exp ( - x/b)} \right\}^{ - 1}}\)
Kernel	\(frac{{\exp ( - x/b)}}{{b{{\left( {1 + \exp ( - x/b)} \right)}^2}}}\)
Support	$\left[ -\infty,\infty \right) $
Mean	0. (A non-zero mean is a relatively easy change)
Variance	\({\pi ^2}{b^2}/3\)
Main Uses	Probability in logit model, transition probability mapping in Markov Switching models, transition function in LSTAR models.
Density Function	%LNLOGISTIC(x) is the log density with b=1 (most common case)
CDF Function	%LOGISTIC(x,b) is logistic CDF of x with scale b. You should use this rather than computing directly, as %LOGISTIC does the calculation in an "overflow-safe" fashion. (In the direct calculation, a large negative \(x\) will cause \({\exp ( - x/b)}\) to overflow).