A Discrete expansion of the Lindley distribution: Mathematical and statistical characterizations with estimation techniques, simulation, and goodness-of-fit analysis

The objective of this paper is to introduce the discretized two-parameter Lindley (D2PL) distribution, a novel discrete probability model that extends the classical Lindley distribution into the discrete domain. This distribution features two parameters, providing greater modeling flexibility and encompassing existing discrete models, such as the one-parameter discrete Lindley and geometric distributions. The paper thoroughly characterizes the D2PL distribution, deriving several key properties essential for reliability modeling. Additional analyses include infinite divisibility, log-convexity, and classical moment measures such as raw moments, dispersion index, skewness, and kurtosis, offering insights into the distribution's shape and tail behavior. The probability mass function of D2PLD can exhibit unimodal and decreasing forms, making it useful for asymmetric count data. Its hazard rate function can model various failure rate patterns, accommodating both under-dispersed and over-dispersed count data. Parameter estimation is conducted through maximum likelihood and method of moments, with Monte Carlo simulations verifying the efficiency and reliability of the estimators. The model's robustness is further demonstrated through applications on real-world count datasets, showing superior goodness of fit over established discrete distributions, highlighting its effectiveness for complex discrete data.