Once the model parameter vector that minimizes the least-squares fit is found, the standard errors in the extracted parameter set can be estimated by computing:
where 
is the constant variance of the measurement errors.
is an estimate of the parameter covariance matrix at the
solution. The diagonal elements of this matrix are the variances
(square of the standard deviation) of the fitted parameters. The
off-diagonal terms 
are the covariances between the 
th and
th parameter. The variance in measurements is usually
approximated as:
where 
is the value of the objective function at the solution.
Details on the statistical theory behind the derivation of (2.35)
can be found in [93][84][7][6].
An important assumption in the derivation is that the mathematical model
can be approximated by a linear function of the parameters near the minimum
and that the measurement errors obey a normal distribution.
For nonlinear models, failure of the first assumption leads to an
unrealistically larger estimate of the errors [7].
The estimate of the standard deviation of the parameter values may be used to calculate approximate confidence intervals for the parameters. For example, the 95% confidence region is:
Monte Carlo simulation techniques can also be used to estimate the errors
in the model parameters [84].
