In this paper, alpha power transformation for continuous distributions is adapted to discrete distributions. The new family for discrete distributions called discrete alpha power transformation is proposed. The discrete alpha power transformation-exponential distribution is studied in detail. Several distributional properties of introduced distribution including moments, survival and hazard rate functions, mode, and quantile function are discussed. The statistical inference on the model parameters is studied by maximum likelihood, moments, and least-squares estimation methods. A simulation study is performed to observe the performance of bias and mean square errors of these estimates. Three bootstrap methods are considered for constructing confidence intervals for the distribution parameters. As an application of the discrete alpha power transformation-exponential distribution, a new zero-inflated count regression model is proposed to be an alternative model for zero-inflated Poisson, zero-inflated geometric, and zero-inflated negative binomial regression models. Two examples with real data are provided to illustrate the applicability of introduced distribution and count regression analysis.

在本文中，连续分布的 alpha 功率变换适用于离散分布。提出了称为离散α功率变换的离散分布的新族。详细研究了离散α功率变换-指数分布。讨论了引入分布的几个分布特性，包括矩、生存和危险率函数、模式和分位数函数。通过最大似然、矩和最小二乘估计方法研究模型参数的统计推断。进行模拟研究以观察这些估计的偏差和均方误差的表现。考虑了三种引导方法来构建分布参数的置信区间。作为离散α幂变换-指数分布的应用，提出了一种新的零膨胀计数回归模型，作为零膨胀泊松、零膨胀几何和零膨胀负二项式回归模型的替代模型。提供了两个真实数据的例子来说明引入分布和计数回归分析的适用性。