New closed-form estimator and its properties

Abstract

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.