A novel two-parameter discrete exponentiated exponential distribution with its neutrosophic extension: mathematical characterization and real-world count data modeling.

This paper proposes and studies the Discrete Exponentiated Exponential (DEE) distribution, a new two-parameter discrete lifetime model obtained by applying the survival discretization technique to the exponentiated exponential distribution. The DEE distribution possesses a key structural advantage: the single shape parameter α simultaneously controls both the hazard rate shape (increasing for 0 < α < 1, decreasing for α > 1) and the dispersion regime (over-dispersion for small α, under-dispersion for large α), a dual flexibility that most competing discrete models cannot replicate without structural modification. These properties are rigorously established in closed form via the Glaser technique and numerical investigation of the dispersion index. Closed-form expressions are derived for the probability mass function, cumulative distribution function, survival and hazard rate functions, moment generating and characteristic functions, raw and central moments up to the fourth order, and the mean residual life function. Parameter estimation is carried out via maximum likelihood, and a comprehensive Monte Carlo simulation study assesses the finite-sample performance of the estimators and benchmarks it against four competing discrete models under identical settings. The practical advantage of the DEE distribution is demonstrated through three real-world count datasets from oncology, education, and nephrology; the DEE model exhibits competitive and frequently superior performance relative to seven benchmark discrete distributions. Finally, a neutrosophic generalization (DNEE) extends the framework to count data characterized by vagueness, incompleteness, or indeterminacy, with accompanying estimation and illustrative numerical analysis.