A Probability Generating Function Based Goodness-of-Fit Test for the Poisson–Three-Parameter Lindley Distribution

The Poisson–Three-Parameter Lindley (PTPL) distribution constitutes a flexible Poisson mixture model for overdispersed count data, encompassing several classical count distributions as special or limiting cases. Despite its growing use in applied contexts, no formal goodness-of-fit test specifically designed for this distribution is currently available. In this paper, we propose and study a new goodness-of-fit test for the PTPL model based on a Cramér–von Mises type distance between the empirical and theoretical probability generating functions (PGFs). For polynomial weight functions, the test statistic admits an explicit closed-form representation; in practice, it is computed efficiently via numerical quadrature. The null distribution of the statistic is approximated via parametric bootstrap. We establish theoretical properties of the proposed procedure, including consistency against fixed alternatives and the validity of the bootstrap approximation. Monte Carlo simulations with sample sizes n∈{50, 100, 150, 200, 500} for size evaluation and n∈{100, 250, 500} for power comparisons, as well as weight exponents a∈{0, 1, 2}, show that the empirical size is well controlled at both the 5% and 10% nominal levels, and that the test exhibits competitive power against Poisson, Negative Binomial, COM-Poisson, and Zero-Inflated Poisson alternatives. A real data application to five overdispersed count datasets further illustrates the practical utility of the method. The empirical size is further verified across twelve parameter configurations spanning dispersion indices from 1.37 to 59.33, confirming bootstrap validity under strong overdispersion.