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In this paper, we refined the concept of past extropy measure for concomitants of order statistics from Farlie-Gumbel-Morgenstern family. In addition, cumulative past extropy measure and dynamic cumulative past extropy measure for concomitant of rth order statistic are also conferred and their properties are studied. The problem of estimating the cumulative past extropy is investigated using empirical technique. The validity of the proposed estimator has been emphasized using simulation study.

Count data are encountered in real-life dealings. More understanding of such data and the extraction of important information about the data require some statistical analysis or modeling. One innovative technique to increase the modeling flexibility of well-known distributions is to use the convolution of random variables. This study examines the distribution that results from adding two independent random variables, one with the Bernoulli distribution and the other with the Poisson–Lindley distribution. The considered distribution is named as the two-parameter Bernoulli–Poisson–Lindley distribution. Many of its statistical properties are investigated, such as moments, survival and hazard rate functions, mode, dispersion behavior, mean deviation about the mean, and parameter inference based on the maximum likelihood method. To evaluate the effectiveness of the bias and mean square error of the produced estimates, a simulation exercise is carried out. Then, applications to two practical data sets are given. Finally, we construct a flexible count data regression model based on the proposed distribution with two practical examples.

Several pieces of research have spotlighted the importance of count data modelling and its applications in real-world phenomena. In light of this, a novel two-parameter compound-Poisson distribution is developed in this paper. Its mathematical functionalities are investigated. The two unknown parameters are estimated using both maximum likelihood and Bayesian approaches. We also offer a parametric regression model for the count datasets based on the proposed distribution. Furthermore, the first-order integer-valued autoregressive process, or INAR(1) process, is also used to demonstrate the utility of the suggested distribution in time series analysis. The unknown parameters of the proposed INAR(1) model are estimated using the conditional maximum likelihood, conditional least squares, and Yule–Walker techniques. Simulation studies for the suggested distribution and the INAR(1) model based on this innovative distribution are also undertaken as an assessment of the long-term performance of the estimators. Finally, we utilized three real datasets to depict the new model’s real-world applicability.