PAPP103 - S07: Demographic models: stable population models

Method of Moments

Theory

An alternative method of calculating r is the method of moments. This method depends on calculating certain characteristics of the reproductivity schedule: its mean and variance. We will not examine the mathematics in detail.

The formula used is:

$r = \frac{\frac{R_{1}}{R_{0}} - \sqrt{{(\frac{R_{1}}{R_{0}})}^{2} - 2 {\frac{R_{2}}{R_{0}} - {(\frac{R_{1}}{R_{0}})}^{2} \ln R_{0}}}}{\frac{R_{2}}{R_{0}} - {(\frac{R_{1}}{R_{0}})}^{2}}$

Method of moments formula for r

Where

$R_{n} = \sum_{10}^{50} x^{n} l_{x} f_{x}$

In other words:

$\begin{array}{l} R_{0} = N R R \\ \frac{R_{1}}{R_{0}} = mean age of f_{x} l_{x} schedule \\ \frac{R_{2}}{R_{0}} - {(\frac{R_{1}}{R_{0}})}^{2} = variance of f_{x} l_{x} schedule \end{array}$

R_n is called the n’th moment of the f_xl_x schedule, hence the name "method of moments".

Estimates of r produced using this method are fairly reliable, especially if evaluated using a single year age distribution of the underlying rates. The method will produce values of Lotka’s balancing equation close to 1.

The worked example on the following page will explain the method.