  We propose new goodness-of-fit tests for the Poisson distribution. The
testing procedure entails fitting a weighted Poisson distribution, which has
the Poisson as a special case, to observed data. Based on sample data, we
calculate an empirical weight function which is compared to its theoretical
counterpart under the Poisson assumption. Weighted Lp distances between these
empirical and theoretical functions are proposed as test statistics and closed
form expressions are derived for L1, L2 and L1 distances. A Monte Carlo study
is included in which the newly proposed tests are shown to be powerful when
compared to existing tests, especially in the case of overdispersed
alternatives. We demonstrate the use of the tests with two practical examples.
