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Variances of the critical point of a quadratic regression equation

The aim of this paper is determine variances for the analysis of the critical point of a second-degree regression equation in experimental situations with different variances through Monte Carlo simulation. In many theoretical or applied studies, one finds situations involving ratios of random variables and more frequently normal variables. Examples are provided by variables, which appear in economic dose research of nutrients in fertilization experiments, as well as in other problems in which there are interests in the random variable<img src="/img/revistas/cagro/v28n2/a20img01.gif">, estimator of the critic point in the regression <img src="/img/revistas/cagro/v28n2/a20img02.gif">. Data of five hundred thirty six trials in cotton yield were utilized to study the distribution of the critical point of a quadratic regression equation by adjusting a quadratic model. The parameters were evaluated using a least square method. From the estimations a MATLAB routine was implemented to simulate two sets with five thousands random errors with normal distribution and zero mean, relative to each of the theoretical variances: <img src="/img/revistas/cagro/v28n2/a20img03.gif" > or = 0.1; 0.5; 1; 5; 10; 15; 20 and 50. The estimation of the variance of the critical point was obtained by three methods: (a) usual formula for the variance; (b) formula obtained by differentiation of the critical point estimator and (c) formula for the computation of the variance of a quotient by taking into consideration the covariance between <img src="/img/revistas/cagro/v28n2/a20img04.gif"> and <img src="/img/revistas/cagro/v28n2/a20img05.gif">. The results obtained for the  statistic  average  for  the  regression between <img src="/img/revistas/cagro/v28n2/a20img04.gif"> e <img src="/img/revistas/cagro/v28n2/a20img05.gif">, as well as its respective variances in terms of the several theoretical residual variances (<img src="/img/revistas/cagro/v28n2/a20img03.gif">) adopted show that those theoretical values are close to real ones. Moreover, there is a trend of increasing <img src="/img/revistas/cagro/v28n2/a20img04.gif"> and <img src="/img/revistas/cagro/v28n2/a20img05.gif"> with increase of the theoretical variance. It may be concluded that the critical point variance calculated taking into consideration the covariance between <img src="/img/revistas/cagro/v28n2/a20img04.gif"> and <img src="/img/revistas/cagro/v28n2/a20img05.gif">, gives more satisfactory results and does not follow a normal distribution, presenting a frequency distribution with positive assimetry and leptokurtic shape.

quadratic regression; quotient random variables; variance of the critical point; interval of confidence


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