To ease a later calculation, we place the result into a matrix structure.
deriv_2nd_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 second partial derivative with respect to \(\theta\),
the second column contains the partial derivative with respect to \(\theta\) and \(\sigma ^2\),
and lastly we have the second partial derivative with respect to \(\sigma ^2\).
Taking the second derivative with respect to \(\theta\) yields: $$\frac{{{\partial ^2}}}{{\partial {\theta ^2}}}\nu _j^2\left( {\theta ,{\sigma ^2}} \right) = \frac{{2{\sigma ^2}}}{{{\tau _j}}}$$
Taking the second derivative with respect to \(\sigma^2\) yields: $$\frac{{{\partial ^2}}}{{\partial {\sigma ^4}}}\nu _j^2\left( {\theta ,{\sigma ^2}} \right) = 0$$
Taking the first derivative with respect to \(\theta\) and \(\sigma^2\) yields: $$\frac{\partial }{{\partial \theta }}\frac{\partial }{{\partial {\sigma ^2}}}\nu _j^2\left( {\theta ,{\sigma ^2}} \right) = \frac{{2(\theta + 1){\tau _j} - 6}}{{\tau _j^2}}$$