Once data have been read in and a model defined, XSPEC uses a modified Levenberg-Marquardt algorithm (based on CURFIT from Bevington, 1969) to find the best-fit values of the model parameters. The algorithm used is a local one, so the user should be aware that it is possible for the fitting process to get stuck in a local minimum and not find the global best-fit. The process also goes much faster if the initial model parameters are set to sensible values.
At the end of a fit, XSPEC will write out the best-fit parameter values, along with estimated confidence intervals. These confidence intervals are one sigma and are calculated from the derivatives of the fit statistic with respect to the model parameters. However, the confidence intervals are not reliable and should be used for indicative purposes only.
XSPEC has separate commands (error or uncertainty) to derive confidence intervals for one interesting parameter, which it does by fixing the parameter of interest at a particular value and fitting for all the other parameters. New values of the parameter of interest are chosen until the appropriate delta-statistic value is obtained. XSPEC uses a bracketing algorithm followed by an iterative cubic interpolation to find the parameter value at each end of the confidence interval.
To compute confidence regions for several parameters at a time XSPEC runs a grid on these parameters. XSPEC also will display a contour plot of the confidence regions of any two parameters.