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:

:$\begin f(m)= \frac \exp\left(\alpha \mu _0+ \frac\right) m^ \text\left( \frac\left(\alpha \sigma _0 -\frac\right)\right),\ m \in [0,\infty) \end$

where $\begin \alpha = \frac \end$ 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 ($F(m) = \int_^m f(t) \,dt$) is given by:

:$\begin F(m) = \frac \text\left(-\frac\right) - \frac \exp\left(\alpha \mu _0 + \frac\right) m^ \text\left(\frac\left(\alpha \sigma _0 - \frac\right)\right) \end$

We can see that as $m\to 0,$ that $\textstyle F(m)\to \frac \operatorname\left(-\frac\right),$ which is the cumulative distribution function for a lognormal distribution with parameters ''μ''₀ and ''σ''₀.

Mean, variance, raw moments

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

:$\begin \langle M^k\rangle = \int _0 ^ m^k f(m) \mathrm dm \end$

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

:$\begin \langle M^k\rangle = \frac \exp\left(\frac + \mu_0 k\right),\ \alpha > k \end$

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

:$\begin \langle M \rangle = \frac \exp\left(\frac + \mu _0\right),\ \alpha > 1 \end$

:$\begin \langle M^2 \rangle = \frac \exp\left(2\left(\sigma ^2 _0 + \mu _0\right)\right),\ \alpha > 2 \end$

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

:$f'(m) = 0 \Leftrightarrow K \operatorname(u) = \exp(-u^2),$

where $\textstyle u = \frac \left( \alpha\sigma_0 - \frac \right)$ and $K = \sigma_0(\alpha+1)\tfrac.$

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:

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

:$\begin E(\delta) = - \delta^ \ln(R(0,1)) \end$
where R(0,1) is the uniform random variate in the interval ,1 Using these two, we can derive the random variate for the MLP distribution to be:

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

References

# {{cite journal, last1=Basu, first1=Shantanu, last2=Gil, first2=M, last3=Auddy, first3=Sayatan, title=The MLP distribution: a modified lognormal power-law model for the stellar initial mass function, journal=MNRAS, date=April 1, 2015, volume=449, issue=3, pages=2413–2420, doi=10.1093/mnras/stv445, arxiv=1503.00023, bibcode=2015MNRAS.449.2413B

Normal distribution