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I am trying to understand the mathematics behind Ridge regression for parametric GAMLSS. As I understand so far, it introduces a penalty term defines as $$\text{Penalty term}=\lambda \sum_{j=1}^{J_k} \beta_j^k$$.

My question is then, when doing Ridge regression for parametric GAMLSS (Generalized additive models for location scale and shape) is the term then substracted from the likelihood or the log-likelihood? I have problems fiding souces that specify this.

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Ridge regression ri() is an additive term that can be included in the predictor for a distribution parameter.

Hence it is fitted locally, using local partial residuals e and local weights w. So the penalty term is added to the local wighted quadratic (e-Xb)'W(e-Xb).

[So the penalty is not added to the global log likelihood.]

For reference see: Stasinopoulos et al. (2017) pages 68-69 and pages 282-287.

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  • $\begingroup$ Thanks for the detail. Do you have a source for this? I have trouble finding literature going into detail with this :) $\endgroup$ Jul 5, 2023 at 20:01

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