deriv_2nd_ma1 {simts} | R Documentation |
Analytic second derivative for MA(1) process
Description
To ease a later calculation, we place the result into a matrix structure.
Usage
deriv_2nd_ma1(theta, sigma2, tau)
Arguments
theta |
A |
sigma2 |
A |
tau |
A |
Value
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
.
Process Haar WV Second Derivative
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}}
Author(s)
James Joseph Balamuta (JJB)