The log-likelihood and its analytic gradient with respect to
(μ, ω, α, γ, β); with scores, each observation's gradient, a row per
parameter.
With L = ln s and k = α·sign(z) + γ, the recursion's derivative is
dL_t = ∂direct + (β − ½·k·z{t−1})·dL_{t−1} + k·e^{−L_{t−1}/2}·de_{t−1}
(de/dμ = −1), and dl/dL = −½(1 − e²/s).
The log-likelihood and its analytic gradient with respect to (μ, ω, α, γ, β); with
scores, each observation's gradient, a row per parameter.With L = ln s and k = α·sign(z) + γ, the recursion's derivative is dL_t = ∂direct + (β − ½·k·z{t−1})·dL_{t−1} + k·e^{−L_{t−1}/2}·de_{t−1} (de/dμ = −1), and dl/dL = −½(1 − e²/s).