Variâncias do ponto critico de equações de regressão quadrática

Abstract

The present work was intended to determine variances for the study ofthe criticai point of a second-degree regression equation under experimental situations with different variances by means of Monte Cario simulation. In a number of studies, whether theoretical or applied, the researcher faces the problem, involving quotient among random variables and mainly among normal variables As an example, those ones which appear in research ofeconomic dose of nutriente in fertilization experimente, in soil compaction and in other problems in which there are interests in the random variable x=b/(-2c), estimator ofthe critic point in the regression y =â +bx +cx2. To study the distribution ofthe criticai point ofa quadratic regression equation, data offive hundred and thirty -sixtrials in cotton yield by adjusting a quadratic model were utilized. From these estimates, a routine for the simulating oftwo sete with five thousand random errors ofnormal distribution ofzero mean relative to each of the variances considered theoretical: <r2=0,l; 0,5; 1; 5; 10; 15; 20 and 50 was implemented by means ofthe MATLAB® software The estimates of the variance ofthe criticai point were obtained through three methods: (a) common formula of the variance calculation; (b) formula obtained through the differentiation ofthe criticai point estimator and (c) formula demonstrated for the variance calculation of a ratio, by taking into consideration the covariance between b e c. The resulte obtained forthe average statistics of the regression between b e c, as well as ite respective variances in terms ofthe several theoretical residual variances (cr2) adopted, show that those theoretical values are close to the real ones. Still, a trend occurs that with the increase of the theoretical variance those values increase. It can conclude thatthe criticai point variance calculated byusing the expression, which takes into consideration the covariance between b e c presente more satisractory resulte and that does not follow a normal distribution, for it presente a frequency distribution with positive asymmetry andleptokurtic shape.