A Bayesian Hidden Markov Model with Singh–Maddala Emission Distributions for Heavy-Tailed Regime-Switching Time Series
Safaa K. Kadhem
Abstract
Heavy-tailed time series that exhibit regime switching are
common in environmental, financial, and engineering
applications, yet their statistical modeling remains
challenging because conventional emission distributions
--- such as the Gaussian, Gamma, or Weibull --- fail to
capture simultaneously the extreme right-skewness, the
slowly decaying upper tails, and the temporal dependence
that characterize such processes.
This paper develops a fully Bayesian Hidden Markov Model
with Singh--Maddala (Burr Type~XII) emission
distributions (HMMSM). We derive the complete Bayesian
structure of the model --- joint prior, complete-data
likelihood, marginal likelihood via the forward
recursion, joint posterior, and full conditional
distributions --- and design an adaptive
Metropolis-within-Gibbs sampler that combines exact
Forward-Filtering Backward-Sampling for the latent
states, conjugate Dirichlet updates for the transition
structure, and adaptive Metropolis steps on the
unconstrained log-scale for the emission parameters.
Extensive simulation studies confirm parameter recovery
with relative errors below $10\%$ and a $95\%$
prediction-interval coverage of $94.5\%$. On a simulated
benchmark with heavy tails ($\alpha_3 = 2.25$), Bayesian
model selection via WAIC, DIC, and LOO correctly
identifies $K = 3$, and HMMSM outperforms seven competing
specifications by large differences
($\Delta\text{WAIC} \geq 51.76$). On a real 26-year
gridded rainfall record for the Republic of Ireland
($T = 312$ monthly observations, 2000--2025), HMMSM
attains the lowest WAIC among all candidates but is
statistically indistinguishable from HMM-Gamma
($\Delta\text{WAIC} = 1.04$), reflecting the relatively
light tails of Irish monthly rainfall
($\hat{\alpha}_1 = 9.60$, $\hat{\alpha}_2 = 10.21$). The
selected two-regime model identifies a Moderate regime
(conditional mean $\approx 87$~mm) and a Wet regime
($\approx 134$~mm) that are persistent and seasonally
coherent, with the Wet regime dominating autumn and
winter. We also document a fundamental weak
identifiability of the Singh--Maddala scale parameter in
this application and recommend that applied users report
inference on the identifiable parameters as their
primary targets.
The framework is general-purpose and applicable to any
positive-valued process with heavy tails and latent
regime dynamics. Open-source, reproducible code is provided
for reproducibility.