autocorrelated data; unit-Lindley distribution; control chart; conditional maximum likelihood; rates and proportions

In this work, new statistical models are developed for the analysis of data that exhibit variation over time. In particular, when the variable (or characteristic) of interest is continuous in the interval (0, 1), as is the case, for example, with rates, proportions and indices. Among the probability distributions in the unit interval that have been introduced in recent literature and have interesting and useful properties (e.g., a single parameter, a reparameterized version in terms of the mean, closed-form expressions for moments), the unit-Lindley distribution stands out. In this work, we propose the unit-Lindley autoregressive and moving average (ULARMA) model, as an extension of the unit-Lindley distribution for the case of autocorrelated data. Furthermore, to control future observations of the process, new control charts are also presented for monitoring and forecasting data of this type. Numerical simulation studies are carried out to evaluate the performance of estimation procedures (e.g., based on the conditional maximum likelihood method) and control charts (e.g., based on the time series model with a continuous response variable at (0, 1) and described by the unit-Lindley distribution) proposed. Finally, the methodology developed here is illustrated in a real data set with information on maximum and minimum values of daily air relative humidity, in the Atacama Desert, located in the north of Chile, in order to verify its applicability in a practical context, when compared with traditional/existing techniques.