Hyperparameter Estimation in Exponential Family State Space Models

Data-driven hyperparameter estimation or automatic choice of the
smoothing parameter is of great importance, especially in the
applications. This article presents and compares three methods for
hyperparameter estimation in the framework of exponential family
state space models: First, we motivate and derive a formula for an
approximative likelihood, and an alternative, yet mathematical
equivalent, expression proves to be a generalized version of a
proposal in Durbin and Koopman (1992). Second, the EM-type
algorithm suggested in Fahrmeir (1992) is restated here for reasons
of comparison and third, the idea of cross-validation proposed by
Kohn and Ansley (1989) for linear state space models is extended to
the present context, in particular for multicategorical and
multidimensional responses. Finally, we compare the three methods
for hyperparameter estimation by applying each on three real data
sets.