deriv_2nd_ar1 {simts}R Documentation

Analytic second derivative matrix for AR(1) process

Description

Calculates the second derivative for the AR(1) process and places it into a matrix form. The matrix form in this case is for convenience of the calculation.

Usage

deriv_2nd_ar1(phi, sigma2, tau)

Arguments

phi

A double corresponding to the phi coefficient of an AR(1) process.

sigma2

A double corresponding to the error term of an AR(1) process.

tau

A vec containing the scales e.g. 2τ2^{\tau}

Value

A matrix with the first column containing the second partial derivative with respect to ϕ\phi and the second column contains the second partial derivative with respect to σ2\sigma ^2

Process Haar WV Second Derivative

Taking the second derivative with respect to ϕ\phi yields:

2ϕ2νj2(ϕ,σ2)=2σ2((ϕ21)τj(2(ϕ(7ϕ+4)+1)ϕτj21(ϕ(7ϕ+4)+1)ϕτj1+3(ϕ+1)2)+(ϕ21)2τj2(ϕτj21)ϕτj21+4(3ϕ+1)(ϕ2+ϕ+1)(ϕτj4ϕτj2+3))(ϕ1)5(ϕ+1)3τj2\frac{{{\partial ^2}}}{{\partial {\phi ^2}}}\nu _j^2\left( \phi, \sigma ^2 \right) = \frac{2 \sigma ^2 \left(\left(\phi ^2-1\right) \tau _j \left(2 (\phi (7 \phi +4)+1) \phi ^{\frac{\tau _j}{2}-1}-(\phi (7 \phi +4)+1) \phi ^{\tau _j-1}+3 (\phi +1)^2\right)+\left(\phi ^2-1\right)^2 \tau _j^2 \left(\phi ^{\frac{\tau _j}{2}}-1\right) \phi ^{\frac{\tau _j}{2}-1}+4 (3 \phi +1) \left(\phi ^2+\phi +1\right) \left(\phi ^{\tau _j}-4 \phi ^{\frac{\tau _j}{2}}+3\right)\right)}{(\phi -1)^5 (\phi +1)^3 \tau _j^2}

Taking the second derivative with respect to σ2\sigma^2 yields:

2σ4νj2(σ2)=0\frac{{{\partial ^2}}}{{\partial {\sigma ^4}}}\nu _j^2\left( \sigma ^2 \right) = 0

Taking the derivative with respect to ϕ\phi and σ2\sigma ^2 yields:

2ϕσ2νj2(ϕ,σ2)=2((ϕ21)τj(ϕτj2ϕτj2ϕ1)(ϕ(3ϕ+2)+1)(ϕτj4ϕτj2+3))(ϕ1)4(ϕ+1)2τj2\frac{{{\partial ^2}}}{{\partial {\phi } \partial {\sigma ^2}}}\nu _j^2\left( \phi, \sigma ^2 \right) = \frac{2 \left(\left(\phi ^2-1\right) \tau _j \left(\phi ^{\tau _j}-2 \phi ^{\frac{\tau _j}{2}}-\phi -1\right)-(\phi (3 \phi +2)+1) \left(\phi ^{\tau _j}-4 \phi ^{\frac{\tau _j}{2}}+3\right)\right)}{(\phi -1)^4 (\phi +1)^2 \tau _j^2}

Author(s)

James Joseph Balamuta (JJB)


[Package simts version 0.2.2 Index]