The language for the log-normal's scale and shape parameters varies depending on the source you check. This app's use of 'scale' and 'shape' is consistent with Box-Steffensmeier and Jones' (2004) general usage of parameter names throughout the book.\( ^{[1]} \)
Where do parameter names come from, in general?
The names we give different parameters stems from how the parameters
modify \( t\)
when you look at a given survivor function's expression. Marshall and Olkin (2007, 218, 612-619) provide a mathematical statement of some possible categories. These categories are not universal, but are nonetheless helpful for seeing the different possibilities.\(^{[2]} \)
To make these determinations for the log-normal, we begin by remembering the log-normal's survivor function:
\( S(t)= \left[ 1-\Phi \left( \frac{\ln(t)-\mu}{\sigma} \right) \right] \)
Marshall and Olkin (2007, 428) take advantage of various log rules to equivalently write the log-normal's survivor as:
\( S(t)= \left[ 1-\Phi \left( \frac{\ln(t)-\ln(\lambda)}{\sigma} \right) \right] \), since \( \mu = X\beta \) and \( \lambda = \exp(X\beta) \), implying \( \mu = \ln(\lambda).\)
\( S(t)= \left[ 1-\Phi \left( \frac{ \ln( \frac{t}{\lambda})}{\sigma} \right) \right] \)
\( S(t)= \left[ 1-\Phi \left( p\ln( \frac{t}{\lambda}) \right) \right] \), since \( p = 1/\sigma \)
\( S(t)= \left[ 1-\Phi \left( \ln \left( \left[ {\frac{t}{\lambda}} \right] ^p \right) \right) \right] \)
This manipulated expression is useful because it allows us to see the relationship between \( t \), \( \mu \) (via \( \lambda \)), and \( \sigma \) (via \( p \)) before \( t \) is logged.
The Verdict
\( \mu \) is the log-normal's
scale
parameter (technically, \( \lambda \)). A parameter \( \theta \) is a scale parameter if it is multiplying or dividing \( t \) by that parameter's value. Formally, scale parameters are of the form:
\( t*\theta \)
\( \mu \) satisfies this rule. In the log-normal's manipulated expression, \( \mu \), in the form of \( \lambda = \exp(\mu) \), appears in the fraction's denominator, when the fraction's numerator contains \( t \): \( \left[ {\frac{t}{\lambda}} \right] \).
\( \sigma \) is what Marshall and Olkin call the log-normal's
power
parameter, but in the language of Box-Steffensmeier and Jones, \( \sigma \) is a
shape
parameter. A parameter \( \theta \) is a shape parameter if \( t \) is being exponentiated by that parameter's value. Formally, shape parameters are of the form:
\( t^\theta \)
\( \sigma \) appears with this general form. In the log-normal's manipulated expression, \( p=\sigma^{-1}\) is the power to which \( t \) is being raised: \( \left[ {\frac{t}{\lambda}} \right] ^p \).
1: Note: Box-Steffensmeier and Jones' *general usage* of parameter names does differ from what they *specifically call* the log-normal's parameters.
2: Marshall and Olkin's categorizations apply to any probability distribution. We've inserted the duration-specific language and notation here, since we're speaking of duration models. Their categorizations refer to the distribution function, \(F(x) \equiv F(t) \), which we know is related to the survivor: \( S(t) = 1 - F(t) \).