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This paper proposes a previously unconsidered generalization of the Lindley distribution by allowing for a measure of noncentrality. Essential structural characteristics are investigated and derived in explicit and tractable forms, and the estimability of the model is illustrated via the fit of this developed model to real data. Subsequently, this model is used as a candidate for the parameter of a Poisson model, which allows for departure from the usual equidispersion restriction that the Poisson offers when modelling count data. This Poisson-noncentral Lindley is also systematically investigated and characteristics are derived. The value of this count model is illustrated and implemented as the count error distribution in an integer autoregressive environment, and juxtaposed against other popular models. The effect of the systematically-induced noncentrality parameter is illustrated and paves the way for future flexible modelling not only as a standalone contender in continuous Lindley-type scenarios but also in discrete and discrete time series scenarios when the often-encountered equidispersed assumption is not adhered to in practical data environments.

A Noncentral Lindley Construction Illustrated in an INAR(1) Environment