On the discrete analogue of the Teissier distribution and its associated INAR(1) process

The continuous Teissier distribution was proposed by the French biologist Georges Teissier in 1934. This paper introduces the one-parameter discrete analogue distribution and studies some of its statistical properties, focussing the attention on the computer generation of pseudo-random data and the parameter estimation problem. More precisely, the new discrete distribution is suitable to deal with both underdispersed and overdispersed count data, the quantile function can be expressed in closed form in terms of the Lambert W function and the failure rate function is increasing. Monte Carlo simulation experiments revealed that estimation methods such as maximum likelihood, least squares and quantile least squares may produce estimates far away from the true parameter value. This drawback can be overcome due to the existence of an analytical expression for the quantile function and then accurate estimates are obtained by the maximum likelihood method even for small sample sizes. Additionally, the new distribution is also used to derive a first-order integer-valued autoregressive process. Different real data sets are used to illustrate that the proposed distribution provides a better fit than other alternative models and also outperforms other competitive processes when time series of counts are considered.