  The question of testing for equality in distribution between two linear
models, each consisting of sums of distinct discrete independent random
variables with unequal numbers of observations, has emerged from the biological
research. In this case, the computation of classical $\chi^2$ statistics, which
would not include all observations, results in loss of power, especially when
sample sizes are small. Here, as an alternative that uses all data, the
nonparametric maximum likelihood estimator for the distribution of sum of
discrete and independent random variables, which we call the convolution
statistic, is proposed and its limiting normal covariance matrix determined. To
challenge null hypotheses about the distribution of this sum, the generalized
Wald's method is applied to define a testing statistic whose distribution is
asymptotic to a $\chi^2$ with as many degrees of freedom as the rank of such
covariance matrix. Rank analysis also reveals a connection with the roots of
the probability generating functions associated to the addend variables of the
linear models. A simulation study is performed to compare the convolution test
with Pearson's $\chi^2$, and to provide usage guidelines.
