This paper introduces a family of ``generalized long-memory time
series models'', in which observations have a specified conditional
distribution, given a latent Gaussian
fractionally integrated autoregressive moving average (ARFIMA) process.
The observations may have discrete or continuous distributions
(or a mixture of both).
The family includes existing models such as ARFIMA models themselves,
long-memory stochastic volatility models, long-memory censored
Gaussian models, and others. Although the family of models
is flexible, the latent long-memory process poses
problems for analysis. Therefore we introduce a
Markov chain Monte Carlo sampling algorithm and develop a
set of recursions which make it feasible.
This makes it possible, among other things,
to carry out exact likelihood-based
analysis of a wide range of non-Gaussian long-memory models
without resorting to the use of likelihood approximations.
The procedure
also yields predictive distributions that take into
account model parameter uncertainty.
The approach is demonstrated in two case studies.