Fractional Lindley distribution generated by time scale theory, with application to discrete-time lifetime data

ABSTRACT

The fractional Lindley distribution is used to model the distribution of perturbations in count data regressions, which allow for dealing with widely dispersed data. It is obtained from the non-fractional Lindley distribution by replacing the support $\mathbb{T}={\mathbb{R}}^{+}$ by $\mathbb{T}=\mathbb{N}\mathrm{\setminus }\{0\}$ and applying time scale theory, whose ambition is to unify the theories of difference equations and differential equations, integral and differential calculus, and the calculus of finite differences. It thus provides a framework for the study of dynamical systems in discrete-continuous time. Delta moments are discrete-time Laplace transforms of the frequency function of the fractional Lindley distribution. The parameter of the fractional Lindley distribution is estimated by least squares, weighted least squares, maximum likelihood, moments, and proportions. The moment estimator always exists, so that delta moments result from the nabla Laplace transform of the frequency function of the fractional Lindley distribution. The maximum likelihood estimates have the least mean-square errors. The proportion method works satisfactorily only when the mode of the distribution is null and the proportion of zeros is high. A simulation allows for quantifying the mean-square errors associated with the estimators. A count regression based on the fractional Lindley distribution with data on the total number of stays after hospital admission among U.S. residents aged 65 and over shows that the Akaike information criteria is significantly lower than with the uniform Poisson and Poisson regressions.

KEYWORDS:

• Discrete-time Laplace transform
• fractional derivatives
• fractional moments
• Shannon entropy
• simulation
• time scale

JEL CLASSIFICATION:

• 60E05
• 62E15
• 93C70
• 26A33
• 39A12

Acknowledgements

The authors thank two reviewers for their helpful comments.

Author contributions

Dr. Fatemeh Gharari wrote the first draft of the manuscript. All authors contributed to the study conception and design. Dr. Kadir Karakaya and Dr. Yunus Akdoğan ran the algorithm and analyzed the results. All authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Disclosure statement

The authors reported no potential conflict of interest.

Additional information

Funding

The authors reported that there is no funding associated with the work featured in this article.