  Nonlinear models are frequently applied to determine the optimal supply
natural gas to a given residential unit based on economical and technical
factors, or used to fit biochemical and pharmaceutical assay nonlinear data. In
this article we propose PRESS statistics and prediction coefficients for a
class of nonlinear beta regression models, namely $P^2$ statistics. We aim at
using both prediction coefficients and goodness-of-fit measures as a scheme of
model select criteria. In this sense, we introduce for beta regression models
under nonlinearity the use of the model selection criteria based on robust
pseudo-$R^2$ statistics. Monte Carlo simulation results on the finite sample
behavior of both prediction-based model selection criteria $P^2$ and the
pseudo-$R^2$ statistics are provided. Three applications for real data are
presented. The linear application relates to the distribution of natural gas
for home usage in S\~ao Paulo, Brazil. Faced with the economic risk of too
overestimate or to underestimate the distribution of gas has been necessary to
construct prediction limits and to select the best predicted and fitted model
to construct best prediction limits it is the aim of the first application.
Additionally, the two nonlinear applications presented also highlight the
importance of considering both goodness-of-predictive and goodness-of-fit of
the competitive models.
