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New closed-form estimator and its properties. (English) Zbl 1485.62024

Summary: 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.

MSC:

62F10 Point estimation
62E20 Asymptotic distribution theory in statistics

Software:

sampleSelection
PDFBibTeX XMLCite
Full Text: DOI

References:

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