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Distributions of settlement populations are modeled using Pareto distributions, probability density function: \\

$ f(x) = \frac{\alpha x^{\alpha}_m}{x^{\alpha + 1}} $ \\

cumulative density function: \\

$ F(x) = 1 - (\frac{x_m}{x})^\alpha $ \\

expected value: \\

$E = \frac{\alpha x_m}{\alpha - 1}$ \\

where $x_m$ – threshold value, $\alpha$ – shape parameter.

A set of values of population sizes can be expressed as a (sorted) series of realizations of identical independent Pareto distributions: \\

$X = (X_{1:n}, ..., X_{n:n}), \quad i \leq j \rightarrow X_{i:n} \leq X_{j:n}, \quad X_{i:n} \sim P(x_m, \alpha)\forall i \in [n]$ \\

where $n$ – number of observations. The probability density of the biggest value in set ($X_{n:n}$) being $m$: \\

\begin{align*}
&\mathcal{P} = P[X_{n:n} = m\ \wedge \bigwedge_{i=1}^{n-1}X_{i:n}\leq m] \overset{independence}{=} \\ &P[X_{n:n} = m] * P[\bigwedge_{i=1}^{n-1}X_{i:n}\leq m] \overset{independence}{=} \\
&P[X_{n:n} = m] * \prod_{i=1}^{n-1}P[X_{i:n}\leq m] \overset{sameness}{=} \\
&P[X_{n:n} = m] * \prod_{i=1}^{n-1}P[X_{1:n}\leq m] = \\
&P[X_{n:n} = m] * (P[X_{1:n}\leq m])^{n-1} = \\
&f(X_{n:n})*(F(X_{n:n}))^{n-1} = \\
&\frac{\alpha x^{\alpha}_m}{X_{n:n}^{\alpha + 1}}*(1 - (\frac{x_m}{X_{n:n}})^\alpha)^{n-1}
\end{align*}

Putting $\gamma = \frac{x_m}{X_{n:n}}$:

\begin{align*}
    \mathcal{P}(\alpha) = \frac{1}{X_{n:n}}(\alpha \gamma^\alpha(1-\gamma^\alpha)^{n-1})
\end{align*}

Maximum likelihood estimator:

\begin{align*}
    &\hat{\alpha} = \underset{\alpha}{maxarg}\{\mathcal{P}(\alpha)\} = \underset{\alpha}{maxarg}\{ln(\mathcal{P}(\alpha))\} = (ln(\mathcal{P}(\alpha)))'_0 = \\ &(ln(\alpha) +\alpha ln(\gamma) + (n-1)ln(1-\gamma^\alpha))'_0 = \\
    &\left(\frac{1}{\alpha}+ln(\gamma)-(n-1)\frac{1}{1-\gamma^\alpha} ln(\gamma)\gamma^\alpha\right)_0
\end{align*}

The expected overall population is the number of colonies above the threshold population times the expected population of a colony above the threshold divided by the percentage of people who live in those: \\

$\hat{N} = n\hat{E}:p = n\frac{\hat{\alpha} x_m}{\hat{\alpha} - 1}:p$

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