The distribution of points of maximum or minimum of second degree regression equations

Abstract

When studying experiments with fertilizers from the economic point of view, it is rather common to use a quadratic regression equation Y = a X² + b X + c, and to seek its point of maximum X = - b/2c. If the values of Y are supposed to be normally distributed, the same happens with the estimates b and c, both linear functions of the Y's. This shows that for the economical interpretation of trials with fertilizers it is important to know what is the distribution oithe ratio of two normal variates. To study the distribution of x= b/2c the statistics y1 and y2 of FISHER (1930) were used, as well as the third and fourth moments. The results obtained show that in most cases the distribution of X is far from normal.