deriv_2nd_ma1: Analytic second derivative for MA(1) process

In SMAC-Group/gmwm: Generalized Method of Wavelet Moments

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:

d^2/dtheta^2 nu[j]^2 (theta, sigma2) = (2*sigma2)/tau[j]

Taking the second derivative with respect to sigma^2 yields:

d^2/dsigma2^2 nu[j]^2 (theta, sigma2) = 0

Taking the first derivative with respect to theta and sigma^2 yields:

d/dtheta * d/dsigma2 nu[j]^2 (theta, sigma2) = (-6 + 2*(1 + theta)*tau[j])/tau[j]^2

Author(s)

James Joseph Balamuta (JJB)

Examples

deriv_2nd_ma1(.3, 1, 2^(1:5))

SMAC-Group/gmwm documentation built on Sept. 11, 2021, 10:06 a.m.