*Article* **The Binomial–Natural Discrete Lindley Distribution: Properties and Application to Count Data**

**Shakaiba Shafiq 1, Sadaf Khan 1, Waleed Marzouk 2, Jiju Gillariose <sup>3</sup> and Farrukh Jamal 1,\***


**Abstract:** In this paper, a new discrete distribution called Binomial–Natural Discrete Lindley distribution is proposed by compounding the binomial and natural discrete Lindley distributions. Some properties of the distribution are discussed including the moment-generating function, moments and hazard rate function. Estimation of the distribution's parameter is studied by methods of moments, proportions and maximum likelihood. A simulation study is performed to compare the performance of the different estimates in terms of bias and mean square error. SO2 data applications are also presented to see that the new distribution is useful in modeling data.

**Keywords:** discretizing; natural discrete Lindley distribution; over dispersion; maximum likelihood estimation

**Citation:** Shafiq, S.; Khan, S.; Marzouk, W.; Gillariose, J.; Jamal, F. The Binomial–Natural Discrete Lindley Distribution: Properties and Application to Count Data. *Math. Comput. Appl.* **2022**, *27*, 62. https://doi.org/10.3390/ mca27040062

Academic Editor: Sandra Ferreira

Received: 8 June 2022 Accepted: 15 July 2022 Published: 20 July 2022

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