The HPNLMOD Procedure

Least Squares Estimation

Models that are estimated by PROC HPNLMOD can be represented by using the equations

\begin{align*}  \mb{Y} & = \mb{f}(\bbeta ;\mb{z}_1,\cdots ,\mb{z}_ k) + \bepsilon \\ \mr{E}[\bepsilon ] & = \mb{0} \\ \mr{Var}[\bepsilon ] & = \sigma ^2\mb{I} \end{align*}

where

$\mb{Y}$

is the $(n \times 1)$ vector of observed responses

$\mb{f}$

is the nonlinear prediction function of parameters and regressor variables

$\bbeta $

is the vector of model parameters to be estimated

$\mb{z}_1,\cdots ,\mb{z}_ k$

are the $(n \times 1)$ vectors for each of the k regressor variables

$\bepsilon $

is the $(n \times 1)$ vector of residuals

$\sigma ^2$

is the variance of the residuals

In these models, the distribution of the residuals is not specified and the model parameters are estimated using the least squares method. For the standard errors and confidence limits in the ‚ÄúParameterEstimates‚ÄĚ table to apply, the errors are assumed to be homoscedastic, uncorrelated, and have zero mean.