When the methods projection (S-Lang or Python help) and covariance (S-Lang or Python help) are used to estimate confidence intervals for thawed model parameters after a fit, sometimes a hard upper or lower limit will be reached for one or more parameters. This produces the message "WARNING: hard minimum hit for parameter <parameter name>", along with a row of dashes in the appropriate place in the projection or covariance output. (The covariance method can also return a null value for an upper/lower limit when the parameter space at the minimum is non-quadratic for a given parameter. The covariance matrix calculations assume that the parameters follow the Normal distribution. If the parameter space is non-smooth, then the covariance calculations fail and Sherpa returns "-----".)

Example projection output:

#S-Lang
sherpa> proj(); 
#Python
sherpa> proj()                                          
WARNING: hard minimum hit for parameter bpow1.gamma2
WARNING: hard maximum hit for parameter bpow1.gamma2
WARNING: hard minimum hit for parameter bpow1.eb
WARNING: hard maximum hit for parameter bpow1.eb
Dataset               = 1
Confidence Method     = projection
Fitting Method        = neldermead
Statistic             = cstat
projection 1-sigma (68.2689%) bounds:
   Param            Best-Fit  Lower Bound  Upper Bound
   -----            --------  -----------  -----------
   bpow1.gamma1       1.54147   -0.0292891   0.0292709
   bpow1.gamma2       8.10056        -----       -----
   bpow1.eb           9.49083       -----        -----
   bpow1.ampl         0.022806 -0.000378395  0.000383854

This occurs when the parameter bound found by projection or covariance lies outside the hard limit boundary for a model parameter - this could result from an issue with the signal-to-noise of the data, the applicability of the model to the data, systematic errors in the data, among others things.

A parameter hard limit represents either a hard physical limit (e.g., temperature is not allowed to go below zero), a mathematical limit (e.g., prevent a number from going to zero or below, when the log of that number will be taken), or the limit of what a float or double can hold (the fit should not be driven above or below the maximum or minimum values a variable can hold). For this reason, model parameter hard limits should not be changed by the user.