SERROR Algorithm

The confidence regions are evaluated by FIT_PARCON. Each parameter in turn is adjusted (using initial guesses LE,UE which are either picked up from the model file or generated from the parameter range) and frozen. FIT_MIN is then called to evaluate the optimum fit obtainable by allowing the other parameters to vary. From the actual fitstat values attained, the parameter offset giving the specified chi-sq. increase (SOFF) is estimated on the assumption that the minimum fitstat surface is a parabolic function of the parameter whose error is being evaluated. The parabola may have a different shape on each side of the minimum, the curve on each side being defined by the assumption of zero slope at the minimum, and the requirement that it should pass through the single offset point evaluated. e.g.

. ! * ! . . ! .<----------------------->!<--------->. - . ! * ^ . ! . SOFF . ! . ^ M ^

where * are the evaluated points, and the resulting confidence limits are shown by <---->.

From Lampton, Margon & Bowyer (Ap.J. 208, p177, 1976) a value SOFF=1 gives a 1 sigma (i.e. 68% confidence) interval and SOFF=2.71 a 90% confidence interval for the case where the model is linear (i.e. the predicted values are linear functions of the parameters). Where this is not the case (i.e. always!) the results may still be approximately right (see LMB and also Avni Ap.J. 210, p642, 1976). For likelihood fitting the Cash statistic C=-2*log(P/P_min) is chi-squared distributed about its minimum (see Cash, Ap.J. 228, 939 (1979)). A safe limit in all cases is obtained by projecting the full NPAR-dimensional confidence interval onto the subspace desired (a single parameter axis in this case). This corresponds to taking SOFF = chi-squared with NPAR d.o.f.( n% conf) for a safe upper bound on the n% confidence limit. For large NPAR this will be way above the linear approximation result.

If the number of free parameters (apart from that whose interval is being evaluated) is reduced as a result of parameter bounds being encountered, then a warning is issued. The confidence region for an affected parameter must be treated with caution. Pegging of a parameter is likely to truncate the confidence limit (since an extra impediment to fstat minimisation is introduced). Where the limit encountered is real (i.e. the user KNOWS it cannot be crossed) the derived confidence limit stands. If the bound is arbitrary then it should be removed and the confidence region reevaluated.

Note that the assumption that the chi-squared surface is parabolic about the best fit may NOT be a good one if the minimum is up against a bound (the minimum of the parabola may then be BEYOND this bound). In this case the initial estimate produced by SERROR can be quite poor, but should converge to the correct value on further invocations.