# Alpha-Power Transformed Lindley Distribution: Properties and Associated Inference with Application to Earthquake Data

## Abstract

The Lindley distribution has been generalized by many authors in recent years. A new two-parameter distribution with decreasing failure rate is introduced, called Alpha Power Transformed Lindley (APTL, in short, henceforth) distribution that provides better fits than the Lindley distribution and some of its known generalizations. The new model includes the Lindley distribution as a special case. Various properties of the proposed distribution, including explicit expressions for the ordinary moments, incomplete and conditional moments, mean residual lifetime, mean deviations, L-moments, moment generating function, cumulant generating function, characteristic function, Bonferroni and Lorenz curves, entropies, stress-strength reliability, stochastic ordering, statistics and distribution of sums, differences, ratios and products are derived. The new distribution can have decreasing increasing, and upside-down bathtub failure rates function depending on its parameters. The model parameters are obtained by the method of maximum likelihood estimation. Also, we obtain the confidence intervals of the model parameters. A simulation study is carried out to examine the bias and mean squared error of the maximum likelihood estimators of the parameters. Finally, two data sets have been analyzed to show how the proposed models work in practice.

## Keywords

Lindley distribution Moments Stress-strength reliability Maximum likelihood estimation## Mathematics Subject Classification

60E05 62F10## Notes

### Acknowledgements

The authors would like to thank the Editor, Associate Editor and anonymous Referee for careful reading and for comments which greatly improved the paper.

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