Modified lognormal power-law distribution

This article may require cleanup to meet Wikipedia's quality standards. The specific problem is: Formatting of mathematical formulas, particularly inappropriate mixing of wikicode and LaTeX. Please help improve this article if you can. (March 2018) (Learn how and when to remove this message)

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.

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

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

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

Following are the few mathematical properties of the MLP distribution:

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

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

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

The expectation value of ${\displaystyle M}$^k gives the ${\displaystyle k}$^th raw moment of ${\displaystyle M}$,

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

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

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

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

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

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

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

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

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

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

Numerical methods are required to solve this transcendental equation. However, noting that if ${\displaystyle K}$≈1 then u = 0 gives us the mode ${\displaystyle m}$^*:

${\displaystyle m^{*}=\exp(\mu _{0}+\alpha \sigma _{0}^{2})}$

The lognormal random variate is:

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

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

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

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

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

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.