Point estimation of the parameter

The message length expression is too difficult to minimise for $\theta$analytically due to the complexities of $z(\theta)$. Nonetheless, it can be done via a combination of analytical and numerical methods.

Equation () can be symbolically differentiate with respect to $\theta$. This results in an equation that has to solved numerically.

$$\begin{split} 0 &= -\frac{h_\theta(\theta)}{h(\theta)} - \fr... ...) + \frac{z(\theta)z_\theta(\theta)}{960} G(\theta) \end{split}$$ (41)

In this case, the subscript $\theta$ indicates that the function has been differentiated with respect to $\theta$. That is $\frac{\partial}{\partial \theta} h(\theta) = h_\theta(\theta)$. This notation is applicable to univariate and multivariate functions.

The expression for $z_\theta(\theta)$ is calculated by symbolically differentiating the expression $z(\theta)$ (equation ) with respect to $\theta$ -

$$\begin{split} z_\theta(\theta) &= \frac{20F(\theta)G_\theta(... ...(\theta)^2 \sqrt{F(\theta)^2+\frac{6}{5}G(\theta)}} \end{split}$$ (42)

However, the expressions for $h_\theta(\theta)$, $h(\theta)$, $f_\theta(x\vert\theta)$, $f(x\vert\theta)$, $F_\theta(\theta)$, $F(\theta)$, $G_\theta(\theta)$ and $G(\theta)$ would depend on the specific problem.