The discrete new XLindley distribution and the associated autoregressive process

Abstract

The continuous new XLindley distribution was introduced by Nawel et al. ( 2023) as a special case of the polynomial exponential distribution proposed by Beghriche et al. (2022). The current paper introduces the one-parameter discrete analogue distribution of the new XLindley model and studies its main statistical properties. In particular, closed-form expressions are provided for the moment generating function, mean, variance, quantile function, hazard rate function and mean residual life. Moreover, the new distribution has discrete increasing failure rate and both overdispersed and underdispersed count data can be handled. The estimation of the unknown parameter can be performed by the maximum likelihood method and a Monte Carlo simulation study reveals that this method provides satisfactory estimates. Additionally, a Ąrst-order integer-valued autoregressive process is constructed from the discrete distribution and, via a simulation study, the conditional maximum likelihood method is recommended for estimation purposes. In order to assess the usefulness in practical applications, the proposed distribution and the associated Ąrst-order autoregressive process are compared to other competing distributions and processes, using to this end several real data sets. In the context of statistical quality control, Ąnally a cumulative sum control chart is developed for monitoring the process mean. To illustrate its usefulness, both simulation and real data analysis are performed.