A generalized Fellner-Schall method for smoothing parameter estimation with application to Tweedie location, scale and shape models

TL;DR

This paper introduces a generalized Fellner-Schall method for estimating smoothing parameters and variance components in regular likelihood models, improving efficiency and applicability, especially for complex models like Tweedie distributions.

Contribution

It extends the Fellner-Schall method to handle multiple smoothing parameters and general Fisher regular likelihoods, simplifying computation and broadening applicability.

Findings

Faster convergence than EM algorithm.

Applicable to complex models like Tweedie distributions.

Simplifies smoothing parameter estimation without sacrificing generality.

Abstract

We consider the estimation of smoothing parameters and variance components in models with a regular log likelihood subject to quadratic penalization of the model coefficients, via a generalization of the method of Fellner (1986) and Schall (1991). In particular: (i) we generalize the original method to the case of penalties that are linear in several smoothing parameters, thereby covering the important cases of tensor product and adaptive smoothers; (ii) we show why the method's steps increase the restricted marginal likelihood of the model, that it tends to converge faster than the EM algorithm, or obvious accelerations of this, and investigate its relation to Newton optimization; (iii) we generalize the method to any Fisher regular likelihood. The method represents a considerable simplification over existing methods of estimating smoothing parameters in the context of regular likelihoods, without sacrificing generality: for example, it is only necessary to compute with the same first and second derivatives of the log-likelihood required for coefficient estimation, and not with the third or fourth order derivatives required by alternative approaches. Examples are provided which would have been impossible or impractical with pre-existing Fellner-Schall methods, along with an example of a Tweedie location, scale and shape model which would be a challenge for alternative methods.