Obtain the first derivative of the MA(1) process.
deriv_ma1(theta, sigma2, tau)A double corresponding to the theta coefficient of an MA(1) process.
A double corresponding to the error term of an MA(1) process.
A vec containing the scales e.g. \(2^{\tau}\)
A matrix with the first column containing the partial derivative with respect to \(\theta\)
and the second column contains the partial derivative with respect to \(\sigma ^2\)
Taking the derivative with respect to \(\theta\) yields: $$\frac{\partial }{{\partial \theta }}\nu _j^2\left( {\theta ,{\sigma ^2}} \right) = \frac{{{\sigma ^2}\left( {2\left( {\theta + 1} \right){\tau _j} - 6} \right)}}{{\tau _j^2}}$$
Taking the derivative with respect to \(\sigma^2\) yields: $$\frac{\partial }{{\partial {\sigma ^2}}}\nu _j^2\left( {\theta ,{\sigma ^2}} \right) = \frac{{{{\left( {\theta + 1} \right)}^2}{\tau _j} - 6\theta }}{{\tau _j^2}}$$