Modified lognormal power-law distribution

The modified lognormal power-law (MLP) function is a three parameter function that can be used to model data that have characteristics of a log-normal distribution and a power law behavior. It has been used to model the functional form of the initial mass function (IMF). Unlike the other functional forms of the IMF, the MLP is a single function with no joining conditions.

Functional form

The closed form of the probability density function of the MLP is as follows:

{\begin{aligned}f(m)={\frac {\alpha }{2}}\exp \left(\alpha \mu _{0}+{\frac {\alpha ^{2}\sigma _{0}^{2}}{2}}\right)m^{-(1+\alpha )}{\text{erfc}}\left({\frac {1}{\sqrt {2}}}\left(\alpha \sigma _{0}-{\frac {\ln(m)-\mu _{0}}{\sigma _{0}}}\right)\right),\ m\in [0,\infty )\end{aligned}}

where ${\begin{aligned}\alpha ={\frac {\delta }{\gamma }}\end{aligned}}$ is the asymptotic power-law index of the distribution. Here $\mu _{0}$ and $\sigma _{0}^{2}$ are the mean and variance, respectively, of an underlying lognormal distribution from which the MLP is derived.

Mathematical properties

Following are the few mathematical properties of the MLP distribution:

Cumulative distribution

The MLP cumulative distribution function ( $F(m)=\int _{-\infty }^{m}f(t)\,dt$ ) is given by:

{\begin{aligned}F(m)={\frac {1}{2}}{\text{erfc}}\left(-{\frac {\ln(m)-\mu _{0}}{{\sqrt {2}}\sigma _{0}}}\right)-{\frac {1}{2}}\exp \left(\alpha \mu _{0}+{\frac {\alpha ^{2}\sigma _{0}^{2}}{2}}\right)m^{-\alpha }{\text{erfc}}\left({\frac {\alpha \sigma _{0}}{\sqrt {2}}}\left(\alpha \sigma _{0}-{\frac {\ln(m)-\mu _{0}}{{\sqrt {2}}\sigma _{0}}}\right)\right)\end{aligned}}

We can see that as $m\to 0,$ that $\textstyle F(m)\to {\frac {1}{2}}\operatorname {erfc} \left(-{\frac {\ln(m-\mu _{0})}{{\sqrt {2}}\sigma _{0}}}\right),$ which is the cumulative distribution function for a lognormal distribution with parameters μ₀ and σ₀.

Mean, variance, raw moments

The expectation value of $M$ ^k gives the $k$ ^th raw moment of $M$ ,

{\begin{aligned}\langle M^{k}\rangle =\int _{0}^{\infty }m^{k}f(m)\mathrm {d} m\end{aligned}}

This exists if and only if α > $k$ , in which case it becomes:

{\begin{aligned}\langle M^{k}\rangle ={\frac {\alpha }{\alpha -k}}\exp \left({\frac {\sigma _{0}^{2}k^{2}}{2}}+\mu _{0}k\right),\ \alpha >k\end{aligned}}

which is the $k$ ^th raw moment of the lognormal distribution with the parameters μ₀ and σ₀ scaled by α⁄α- $k$ in the limit α→∞. This gives the mean and variance of the MLP distribution:

{\begin{aligned}\langle M\rangle ={\frac {\alpha }{\alpha -1}}\exp \left({\frac {\sigma _{0}^{2}}{2}}+\mu _{0}\right),\ \alpha >1\end{aligned}}

{\begin{aligned}\langle M^{2}\rangle ={\frac {\alpha }{\alpha -2}}\exp \left(2\left(\sigma _{0}^{2}+\mu _{0}\right)\right),\ \alpha >2\end{aligned}}

Var( $M$ ) = ⟨ $M$ ²⟩-(⟨ $M$ ⟩)² = α exp(σ₀² + 2μ₀) (⁠exp(σ₀²)/α-2⁠ - ⁠α/(α-2)²⁠), α > 2

Mode

The solution to the equation $f'(m)$ = 0 (equating the slope to zero at the point of maxima) for $m$ gives the mode of the MLP distribution.

f'(m)=0\Leftrightarrow K\operatorname {erfc} (u)=\exp(-u^{2}),

where $\textstyle u={\frac {1}{\sqrt {2}}}\left(\alpha \sigma _{0}-{\frac {\ln m-\mu _{0}}{\sigma _{0}}}\right)$ and $K=\sigma _{0}(\alpha +1){\tfrac {\sqrt {\pi }}{2}}.$