A q -analogue of the Poisson–Lindley distribution: A bridge from the geometric distribution to the Poisson–Lindley distribution

We introduce the q-Poisson–Lindley distribution (q-PLD), a two-parameter count distribution obtained by replacing the negative-binomial count factor in the mixture representation of the Poisson–Lindley distribution (PLD) by its Jackson q-integer. The family has the geometric distribution and PLD as the endpoints of its deformation parameter and retains a fixed geometric tail rate. It is also an exact Jackson q-integral mixture of the Euler distribution over a Lindley-type q-density. We give closed-form expressions for the survival, distribution, hazard, and generating functions and prove log-concavity, the increasing failure rate property, overdispersion, and monotonicity of the mean. The parameters are identifiable, the Poisson–Lindley hypothesis is a boundary hypothesis with a mixed chi-square limit, and a Kullback–Leibler diagnostic shows when the deformation is detectable. For red mite counts, the estimate is at the geometric endpoint; for COVID-19 counts, it lies inside the interval, close to the Poisson–Lindley endpoint; neither supports a practically important interior deformation.