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{align}

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{align}$$

where $$\begin{align} \alpha = \frac{\delta}{\gamma} \end{align}$$ 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{align}

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 ^2 _0}{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{align}$$

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{align}

\langle M^k\rangle = \int _0 ^{\infty} m^k f(m) \mathrm dm \end{align}$$

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


 * $$\begin{align}

\langle M^k\rangle = \frac{\alpha}{\alpha-k} \exp\left(\frac{\sigma_0 ^2 k^2}{2} + \mu_0 k\right),\ \alpha > k \end{align}$$

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{align}

\langle M \rangle = \frac{\alpha}{\alpha-1} \exp\left(\frac{\sigma ^2 _0}{2} + \mu _0\right),\ \alpha > 1 \end{align}$$


 * $$\begin{align}

\langle M^2 \rangle = \frac{\alpha}{\alpha-2} \exp\left(2\left(\sigma ^2 _0 + \mu _0\right)\right),\ \alpha > 2 \end{align}$$

Var($$M$$) = ⟨$$M$$2⟩-(⟨$$M$$⟩)2 = α exp(σ02 + 2μ0) ($exp(σ_{0}^{2})⁄α-2$ - $α⁄(α-2)^{2}$), 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 $$K = \sigma_0(\alpha+1)\tfrac{\sqrt{\pi}}{2}.$$

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


 * $$m^* = \exp (\mu_0+ \alpha \sigma ^2 _0)$$

Random variate
The lognormal random variate is:


 * $$\begin{align}

L(\mu,\sigma) = \exp(\mu+\sigma N(0,1)) \end{align}$$ where $$N(0,1)$$ is standard normal random variate. The exponential random variate is :


 * $$\begin{align}

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

M (\mu_0,\sigma_0,\alpha) = \exp(\mu_0 + \sigma_0 N (0,1) - \alpha^{-1} \ln(R(0,1))) \end{align}$$