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:

$$\begin{gathered} \begin{array}{r}f(m)=\frac{\alpha }{2}\exp (\alpha {\mu }_0+\frac{{\alpha }^2{\sigma }_0^2}{2})m^{-(1+\alpha )}\text{erfc}\left(\frac{1}{\sqrt{2}}(\alpha {\sigma }_0-\frac{\ln (m)-{\mu }_0}{{\sigma }_0})\right), \\ m\in [0,\mathrm{\infty })\end{array} \end{gathered}$$

where $\begin{array}{r}\alpha =\frac{\delta }{\gamma }\end{array}$ 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 ($F(m)=\underset{-\mathrm{\infty }}{\overset{m}{\int }}f(t)dt$) is given by:

$\begin{array}{r}F(m)=\frac{1}{2}\text{erfc}(-\frac{\ln (m)-{\mu }_0}{\sqrt{2}{\sigma }_0})-\frac{1}{2}\exp (\alpha {\mu }_0+\frac{{\alpha }^2{\sigma }_0^2}{2})m^{-\alpha }\text{erfc}\left(\frac{\alpha {\sigma }_0}{\sqrt{2}}(\alpha {\sigma }_0-\frac{\ln (m)-{\mu }_0}{\sqrt{2}{\sigma }_0})\right)\end{array}$

We can see that as $m\to 0,$ that $F(m)\to \frac{1}{2}\mathrm{erfc}(-\frac{\ln (m-{\mu }_0)}{\sqrt{2}{\sigma }_0}),$ 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 $M$,

$\begin{array}{r}⟨M^k⟩=\underset{0}{\overset{\mathrm{\infty }}{\int }}{m}^{k}f(m)\mathrm{d}m\end{array}$

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

$$\begin{gathered} \begin{array}{r}⟨M^k⟩=\frac{\alpha }{\alpha -k}\exp (\frac{{\sigma }_0^2{k}^2}{2}+{\mu }_{0}k), \\ \alpha >k\end{array} \end{gathered}$$

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:

$$\begin{gathered} \begin{array}{r}⟨M⟩=\frac{\alpha }{\alpha -1}\exp (\frac{{\sigma }_0^2}{2}+{\mu }_0), \\ \alpha >1\end{array} \end{gathered}$$

$$\begin{gathered} \begin{array}{r}⟨M^2⟩=\frac{\alpha }{\alpha -2}\exp \left(2({\sigma }_0^2+{\mu }_0)\right), \\ \alpha >2\end{array} \end{gathered}$$

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 $f^{\prime }(m)$ = 0 (equating the slope to zero at the point of maxima) for $m$ gives the mode of the MLP distribution.

$f^{\prime }(m)=0\iff K\mathrm{erfc}(u)=\exp (-u^2),$

where $u=\frac{1}{\sqrt{2}}(\alpha {\sigma }_0-\frac{\ln m-{\mu }_0}{{\sigma }_0})$ and $K={\sigma }_0(\alpha +1)\frac{\sqrt{\pi }}{2}.$ Numerical methods are required to solve this transcendental equation. However, noting that if $K$≈1 then u = 0 gives us the mode $m$^*:

$m^{*}=\exp ({\mu }_0+\alpha {\sigma }_0^2)$

Random variate

The lognormal random variate is:

$\begin{array}{r}L(\mu ,\sigma )=\exp (\mu +\sigma N(0,1))\end{array}$

where $N(0,1)$ is standard normal random variate. The exponential random variate is :

$\begin{array}{r}E(\delta )=-{\delta }^{-1}\ln (R(0,1))\end{array}$

where R(0,1) is the uniform random variate in the interval [0,1]. Using these two, we can derive the random variate for the MLP distribution to be:

$\begin{array}{r}M({\mu }_0,{\sigma }_0,\alpha )=\exp ({\mu }_0+{\sigma }_{0}N(0,1)-{\alpha }^{-1}\ln (R(0,1)))\end{array}$

References

1. Basu, Shantanu; Gil, M; Auddy, Sayatan (April 1, 2015). "The MLP distribution: a modified lognormal power-law model for the stellar initial mass function". MNRAS. 449 (3): 2413–2420. arXiv:1503.00023. Bibcode:2015MNRAS.449.2413B. doi:10.1093/mnras/stv445.