Intervalos de confiança para o ponto crítico de modelos de regressão quadrática: abordagens bootstrap, bayesiana e fuzzy.

Abstract

Using a quadratic regression model where, for example, the production of a certain culture is assessed by different doses of nutrients, might be interesting to obtain a confidence interval for the critical point that represents the value of doses that provide approximately the maximum production culture. This paper has as a main goal, proposes a construction of confidence intervals for the critical point using parametric bootstrap methodology, bayesian inference and fuzzy logic considering data from dry matter production of signal grass at different doses of phosphorus. For the parametric bootstrap analysis were considered different theoretical variances for the error and confidence intervals were constructed according to different expressions of variances of the critical point, beyond the bootstrap-t confidence interval.In bayesian analysis, bayesian confidence intervals (Highest Posterior Density - HPD) were found for the critical point. Thus, priors were defined for each parameter, including the critical point, and through the Gibbs sampler were made inferences. In fuzzy analysis, by the Zadeh’s extension principle, a fuzzy quadratic regression model was found considering uncertainties on estimates of the parameters obtained by the least squares method. Thus, a fuzzy critical point and fuzzy confidence intervals were obtained using interval operations and -cuts. A second fuzzy analysis was performed considering the Buckley’s methodology, in which a fuzzy estimator to the critical point were constructed based on a classical confidence interval. For parametric bootstrap analysis, confidence intervals considered that the expression of the variance, which take into consideration the covariance between the parameters of the regression model showed higher precision, and the frequency distribution of the critical point tends to a positive asymmetric distribution and leptokurtic shape with increasing variance of theoretical. According to the priors assumed, the bayesian confidence intervals found for the critical point showed high precision. Considering the uncertainties treated by the fuzzy analysis, the increased levels of confidence, based on -cuts, resulted in higher precision of the fuzzy confidence intervals. The Buckley’s methodology provided more information than a interval estimate conventional.