  In both Tweedie and geometric Tweedie models, the common power parameter
$p\notin(0,1)$ works as an automatic distribution selection. It mainly
separates two subclasses of semicontinuous ($1<p<2$) and positive continuous
($p\geq 2$) distributions. Our paper centers around exploring diagnostic tools
based on the maximum likelihood ratio test and minimum Kolmogorov-Smirnov
distance methods in order to discriminate very close distributions within each
subclass of these two models according to values of $p$. Grounded on the unique
equality of variation indices, we also discriminate the gamma and geometric
gamma distributions with $p=2$ in Tweedie and geometric Tweedie families,
respectively. Probabilities of correct selection for several combinations of
dispersion parameters, means and sample sizes are examined by simulations. We
thus perform a numerical comparison study to assess the discrimination
procedures in these subclasses of two families. Finally, semicontinuous
($1<p\leq 2$) distributions in the broad sense are significantly more
distinguishable than the over-varied continuous ($p>2$) ones; and two datasets
for illustration purposes are investigated.
