  Time series in real-world applications often have missing observations,
making typical analytical methods unsuitable. One method for dealing with
missing data is the concept of amplitude modulation. While this principle works
with any data, here, missing data for unbounded and bounded count time series
are investigated, where tailor-made dispersion and skewness statistics are used
for model diagnostics. General closed-form asymptotic formulas are derived for
such statistics with only weak assumptions on the underlying process. Moreover,
closed-form formulas are derived for the popular special cases of Poisson and
binomial autoregressive processes, always under the assumption that missingness
occurs. The finite-sample performances of the considered asymptotic
approximations are analyzed with simulations. The practical application of the
corresponding dispersion and skewness tests under missing data is demonstrated
with three real-data examples.
