A first-order integer-valued autoregressive process with one-inflated Borel innovations
One-inflated distributions; INAR(1) processes; Borel distribution
As time series of integer values are based on counts, such as the autoregressive
processes of integer values (INAR), they may contain an excessive number of repeated values
that could harm the inferential analysis if this behavior is not known. Then it becomes
necessary to understand time series inflated models for integer values. Some models have
been developed to study zero-inflated data, however, this work focuses on the one-inflated
model (OI) and the autoregressive process with One-inflated innovations (OI-INAR(1)). Thus,
the One-inflated Borel (OIB) model and the INAR(1) process with One-inflated Borel
innovations (OIB-INAR(1)) are proposed. In this work the properties, such as expectation,
variance, dispersion index and probability generating function, of these models are developed,
as well as the methods for estimating the parameters of the OIB model, such as the method of
moments and maximum likelihood. Likewise, for the OIB-INAR(1) process, the conditional least squares and conditional maximum likelihood methods, in addition to one-step-ahead
prediction methodologies are studied. Finally, an application is made adjusting the OIB-INAR(1)
model.