The problem of statistical modeling of the geometric count data with a specific probability model of lifetimes is of interest and importance in reliability. In this paper, we construct a geometric process (GP), with parameter [Formula: see text], for modeling the geometric count data when the distribution of first occurrence time is a scaled Muth with parameters [Formula: see text] and [Formula: see text]. We investigate the estimators of the process parameters [Formula: see text], [Formula: see text] and [Formula: see text] from a point of approximations of classical and modified approach by using the different estimation methodologies such as the maximum likelihood, moments, least-squares and maximum spacing. We perform a simulation study to compare the estimation performance of the estimators obtained. Finally, we provide an illustrative analysis conducted on a real-world dataset to show the efficiency of the GP model constructed in this paper against the alpha-series and renewal processes and exemplify the data modeling stages. Consequently, a forecasting to such data using the GP with the scaled Muth is investigated.

中文翻译：

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具有比例 Muth 分布的几何过程参数的推断

具有特定寿命概率模型的几何计数数据的统计建模问题在可靠性方面具有重要意义。在本文中，我们构造了一个几何过程（GP），参数为[公式：见文本]，用于在首次出现时间的分布是具有参数[公式：见文本]和[公式]的缩放Muth时对几何计数数据进行建模: 见正文]。我们通过使用不同的估计方法，例如最大似然，从经典和改进方法的近似点研究过程参数的估计量[公式：见文本]、[公式：见文本]和 [公式：见文本]，矩，最小二乘和最大间距。我们进行了一项模拟研究，以比较获得的估计器的估计性能。最后，我们提供了对真实数据集进行的说明性分析，以展示本文构建的 GP 模型对 alpha 系列和更新过程的效率，并举例说明数据建模阶段。因此，研究了使用具有缩放 Muth 的 GP 对此类数据的预测。