In numerous instances, the generalized exponential distribution can be used as an alternative to the most widely used non-regular family of distributions: Weibull, gamma, lognormal with three-parameters when analyzing lifetime or any skewed continuous data. A non-regular family is a class of probability distributions that do not satisfy the regularity conditions typically assumed in classical statistical inference. Some key features of such family of distributions are: support of its probability density function depends on one its parameters; its likelihood function may not be bounded for a certain range of parameter space, hence maximum likelihood estimators do not exist; the likelihood function even may not be differentiable or integrable as needed, hence Fisher Information may not exist or be infinite. Moreover, standard results like MLE existence, consistency, asymptotic normality may fail. Therefore, specialized or robust inferential techniques are needed. This article offers a consistent method for estimating the parameters of a three-parameter generalized exponential distribution that sidesteps the issue of an unbounded likelihood function. The method is hinged on a maximum likelihood estimation of shape and scale parameters that uses a location-invariant statistic. Important estimator properties, such as uniqueness and consistency, are demonstrated for the first time under this approach. In addition, quantile estimates for the assumed distribution are provided. We present a Monte Carlo simulation study along with comparisons to a number of well-known estimation techniques in terms of bias and root mean square error. For illustrative purposes, a real dataset from reliability engineering, has been analyzed and the goodness of fit along with the bootstrap confidence intervals are compared with existing traditional methods.

翻译：在许多情况下，分析寿命数据或任何偏态连续数据时，广义指数分布可作为最广泛使用的非正则分布族（三参数Weibull分布、gamma分布、对数正态分布）的替代选择。非正则分布族是指不满足经典统计推断通常假设的正则条件的一类概率分布。此类分布族的主要特征包括：其概率密度函数的支撑集依赖于某个参数；似然函数在参数空间的特定范围内可能无界，因此极大似然估计量不存在；似然函数甚至可能无法按需进行微分或积分，导致Fisher信息可能不存在或为无穷大。此外，诸如MLE存在性、一致性、渐近正态性等标准结论可能失效。因此需要采用专门或稳健的推断技术。本文提出了一种估计三参数广义指数分布参数的一致性方法，该方法规避了似然函数无界的问题。该方法的核心在于利用位置不变统计量对形状参数和尺度参数进行极大似然估计。在此框架下首次证明了估计量的重要性质，如唯一性和一致性。此外，本文还提供了假定分布的分位数估计。我们通过蒙特卡洛模拟研究，在偏差和均方根误差方面与多种经典估计方法进行了比较。为说明方法的应用，分析了一个可靠性工程领域的真实数据集，并将拟合优度及Bootstrap置信区间与现有传统方法进行了对比。