The Estimation of Non-Linear Parameters by `Internal Least Squares'

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Abstract
If a non-linear regression law may be regarded as a soln. of a linear differential equation, data may be fitted directly to the linear equation, the least squares technic giving linear equations for the parameters. In order to fit an empirical series, it is necessary to use an equivalent difference equation. Summing the difference equation gives an equation for the dependent variable in terms of its own repeated sums and the independent variable. Such an equation is called an "internal regression." The 1st order internal regression is considered for the exponential law of diminishing returns. It is shown that by suitable transformations, the logistic and Gompertz curves may be reduced to exponential form. The asymptotic variances are given for internal least squares estimators and comparisons made with the max. likelihood estimators. The efficiency of the internal regression technic is quite good for moderate curvature of the exponential law.

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