Count data represent the number of occurrences of an event within a fixed time or space and arise frequently in areas such as epidemiology, insurance, demography, and reliability analysis. In many practical situations, zero counts are structurally absent or unobserved, resulting in zero-truncated count data. Standard zero-truncated models may be inadequate when such data exhibit substantial over-dispersion. In this paper, we introduce and study a zero-truncated hyper-negative binomial distribution (ZTHNBD) to addressthis limitation. This work constitutes the first systematic investigation of the ZTHNBD. Several important statistical properties of the proposed distribution are derived, including the probability mass function, cumulative distribution function, mode, log-concavity, survival function, and hazard function.Recurrence relations for probabilities, raw moments, and factorial moments are also obtained. Parameter estimation is carried out using the method of maximum likelihood, and a generalized likelihood ratio test is developed to assess the significance of the additional parameter. The practical usefulness of the proposed model is demonstrated using multiple real-life data sets, where it provides an improved fit compared to existing zero-truncated models based on goodness of fit measures and information criteria. A brief simulation study is conducted to examine the finite sample performance of the maximum likelihood estimators.
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