Least Squares Estimates and Parabolic Regression with Restricted Location for the Stationary Point

Abstract

It is sometimes desired to fit a second degree polynomial E ( y | x ) = α + β x + γ x 2 to a set of data so that its stationary point will fall in a given region. Methods are shown to calculate a, b, c , the least squares estimates of the regression coefficients α, β, and γ. An example is given in which this theory is applied to fit a parabolic regression to a set of data so that in a given contiguous and finite region of the independent variable, the regression function is strictly increasing and concave upwards.