deriv_2nd_ma1: Analytic second derivative for MA(1) process

In SMAC-Group/simts: Time Series Analysis Tools

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 double corresponding to the theta coefficient of an MA(1) process.
sigma2	A double corresponding to the error term of an MA(1) process.
tau	A vec containing the scales e.g. 2^{\tau}

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)

SMAC-Group/simts documentation built on Sept. 4, 2023, 5:25 a.m.