The Geometric–Natural Discrete Lindley Distribution: A One-Parameter Mixture Model with Stochastic Representation for Overdispersed Count Data

This paper studies the geometric–natural discrete Lindley (GNDL) distribution as a structurally motivated one-parameter submodel of the published two-parameter natural discrete Lindley (TNDL) family, rather than as a new unrestricted family. The model arises from a nested geometric–NDL construction and admits equivalent geometric–negative binomial and random-sum representations. Its single parameter jointly controls the common success probability of the geometric stages and the probability of activating the two-stage component, providing parsimony at the cost of preventing these features from varying independently. We derive the main distributional and reliability properties and investigate four frequentist estimators through Monte Carlo simulation. The empirical cost of the one-parameter restriction is assessed against the unrestricted TNDL benchmark in four real-data applications and through a dedicated misspecification study. Across the applications, freeing the second parameter yields negligible log-likelihood gains, while AIC and BIC favor GNDL in every case. Misspecification is inexpensive near the GNDL restriction, whereas larger departures become increasingly detectable as sample size grows. These findings support GNDL as a tractable constrained model when its parameter coupling is compatible with the data, rather than as a universally superior alternative.