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:

[math]\displaystyle{ \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} }[/math]

where [math]\displaystyle{ \begin{align} \alpha = \frac{\delta}{\gamma} \end{align} }[/math] is the asymptotic power-law index of the distribution. Here [math]\displaystyle{ \mu_0 }[/math] and [math]\displaystyle{ \sigma_0^2 }[/math] 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 ([math]\displaystyle{ F(m) = \int_{-\infty}^m f(t) \,dt }[/math]) is given by:

[math]\displaystyle{ \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} }[/math]

We can see that as [math]\displaystyle{ m\to 0, }[/math] that [math]\displaystyle{ \textstyle F(m)\to \frac{1}{2} \operatorname{erfc}\left(-\frac{\ln(m - \mu_0)}{\sqrt{2}\sigma_0}\right), }[/math] which is the cumulative distribution function for a lognormal distribution with parameters μ₀ and σ₀.

Mean, variance, raw moments

The expectation value of [math]\displaystyle{ M }[/math]^k gives the [math]\displaystyle{ k }[/math]^th raw moment of [math]\displaystyle{ M }[/math],

[math]\displaystyle{ \begin{align} \langle M^k\rangle = \int _0 ^{\infty} m^k f(m) \mathrm dm \end{align} }[/math]

This exists if and only if α > [math]\displaystyle{ k }[/math], in which case it becomes:

[math]\displaystyle{ \begin{align} \langle M^k\rangle = \frac{\alpha}{\alpha-k} \exp\left(\frac{\sigma_0 ^2 k^2}{2} + \mu_0 k\right),\ \alpha \gt k \end{align} }[/math]

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

[math]\displaystyle{ \begin{align} \langle M \rangle = \frac{\alpha}{\alpha-1} \exp\left(\frac{\sigma ^2 _0}{2} + \mu _0\right),\ \alpha \gt 1 \end{align} }[/math]

[math]\displaystyle{ \begin{align} \langle M^2 \rangle = \frac{\alpha}{\alpha-2} \exp\left(2\left(\sigma ^2 _0 + \mu _0\right)\right),\ \alpha \gt 2 \end{align} }[/math]

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

Mode

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

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

where [math]\displaystyle{ \textstyle u = \frac{1}{\sqrt{2}} \left( \alpha\sigma_0 - \frac{\ln m - \mu_0}{\sigma_0} \right) }[/math] and [math]\displaystyle{ K = \sigma_0(\alpha+1)\tfrac{\sqrt{\pi}}{2}. }[/math]

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

[math]\displaystyle{ m^* = \exp (\mu_0+ \alpha \sigma ^2 _0) }[/math]

Random variate

The lognormal random variate is:

[math]\displaystyle{ \begin{align} L(\mu,\sigma) = \exp(\mu+\sigma N(0,1)) \end{align} }[/math]

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

[math]\displaystyle{ \begin{align} E(\delta) = - \delta^{-1} \ln(R(0,1)) \end{align} }[/math]

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:

[math]\displaystyle{ \begin{align} M (\mu_0,\sigma_0,\alpha) = \exp(\mu_0 + \sigma_0 N (0,1) - \alpha^{-1} \ln(R(0,1))) \end{align} }[/math]

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. doi:10.1093/mnras/stv445. Bibcode: 2015MNRAS.449.2413B.