## Print version ISSN 1413-7054

### Ciênc. agrotec. vol.28 no.2 Lavras Mar./Apr. 2004

#### http://dx.doi.org/10.1590/S1413-70542004000200020

a20

Variâncias do ponto crítico de equações de regressão quadrática

Variances of the critical point  of a quadratic regression equation

Ceile Cristina Ferreira NunesI; Augusto Ramalho de MoraisII; Joel Augusto MunizIII; Thelma SáfadiII

IMestre em Estatística e Experimentação Agropecuária – Universidade Federal de Lavras/UFLA – Caixa Postal 37 – 37200-000 Lavras, MG
IIProfessores Adjunto do Departamento de Ciências Exatas/UFLA
IIIProfessor Titular do Departamento de Ciências Exatas/UFLA

RESUMO

Termos para indexação: Regressão quadrática, quociente de variáveis aleatórias, variância do ponto crítico, intervalo de confiança.

ABSTRACT

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, estimator of the critic point in the regression . 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: = 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  and . The results obtained for the  statistic  average  for  the  regression between  e , as well as its respective variances in terms of the several theoretical residual variances () adopted show that those theoretical values are close to real ones. Moreover, there is a trend of increasing  and  with increase of the theoretical variance. It may be concluded that the critical point variance calculated taking into consideration the covariance between  and , gives more satisfactory results and does not follow a normal distribution, presenting a frequency distribution with positive assimetry and leptokurtic shape.

Index terms: quadratic regression, quotient random variables, variance of the critical point, interval of confidence.

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(Recebido para publicação em 23 de abril de 2002 e aprovado em 8 de agosto de 2002)