There is no closed form maximum likelihood estimator (MLE) for some distributions. This might cause some problems in real-time processing. Using an extension of Box–Cox transformation, we develop a closed-form estimator for the family of distributions. If such closed-form estimators exist, they have the invariance property like MLE and are equal in distribution with respect to the transformation. Specifically, the joint exact and asymptotic distributions of the closed-form estimators are the same irrespective of the transformation parameter, which is useful for statistical inference. For the gamma related and weighted Lindley related distributions, the closed-form estimators achieve strong consistency and asymptotic normality similar to MLE. That is, the closed-form estimators from the family of distributions obtained from an extension of the Box–Cox transformation for the gamma and weighted Lindley distributions as the initial distributions achieve strong consistency and asymptotic normality. A bias-corrected closed-form estimator that is also independent of the transformation is derived. In this sense, the closed-form estimator and the bias-corrected closed-form estimator are invariant with respect to the transformation. Some examples are provided to demonstrate the underlying theory. Some simulation studies and a real data example for the inverse gamma distribution are presented to illustrate the performance of the proposed estimators in this study.

对于某些分布，没有封闭形式的最大似然估计器（MLE）。这可能会在实时处理中引起一些问题。使用Box-Cox变换的扩展，我们为分布族开发了一种封闭形式的估计器。如果存在这种闭合形式的估计量，则它们具有不变性（如MLE），并且在变换方面的分布是相等的。具体来说，闭合​​形式的估计量的联合精确和渐近分布是相同的，而与变换参数无关，这对于统计推断很有用。对于与伽玛有关和加权与林德利有关的分布，与MLE相似，闭合形式的估计量具有很强的一致性和渐近正态性。那是，从Box-Cox变换对伽玛分布和加权Lindley分布的扩展中获得的分布族的闭合形式估计量，因为初始分布实现了强一致性和渐近正态性。得出也独立于变换的偏差校正后的闭式估计器。从这个意义上讲，闭合形式的估计器和经偏差校正的闭合形式的估计器相对于变换是不变的。提供了一些示例来说明基础理论。提出了一些反伽马分布的仿真研究和一个真实的数据示例，以说明本研究中所提出的估计量的性能。