  We present a novel approach for modeling bounded count time series data, by
deriving accurate upper and lower bounds for the variance of a bounded count
random variable while maintaining a fixed mean. Leveraging these bounds, we
propose semiparametric mean and variance joint (MVJ) models utilizing a
clipped-Laplace link function. These models offer a flexible and feasible
structure for both mean and variance, accommodating various scenarios of
under-dispersion, equi-dispersion, or over-dispersion in bounded time series.
The proposed MVJ models feature a linear mean structure with positive
regression coefficients summing to one and allow for negative regression
cefficients and autocorrelations. We demonstrate that the autocorrelation
structure of MVJ models mirrors that of an autoregressive moving-average (ARMA)
process, provided the proposed clipped-Laplace link functions with nonnegative
regression coefficients summing to one are utilized. We establish conditions
ensuring the stationarity and ergodicity properties of the MVJ process, along
with demonstrating the consistency and asymptotic normality of the conditional
least squares estimators. To aid model selection and diagnostics, we introduce
two model selection criteria and apply two model diagnostics statistics.
Finally, we conduct simulations and real data analyses to investigate the
finite-sample properties of the proposed MVJ models, providing insights into
their efficacy and applicability in practical scenarios.
