Metalog
Proposes candidate bounds for the semi-bounded and bounded metalogs.
Candidates sit at a fraction of the observed range beyond the observed extreme, rather than at the extreme itself. That clearance matters: a bound that merely touches the sample minimum sends the transformed value of that observation to a large negative number, which then dominates the least squares fit. Measured on log-normal samples, a bound taken from the shift estimator that the rest of the fitting subsystem uses agrees with the sample minimum to four or five decimals, and the resulting fit gets steadily worse as terms are added, while a bound with clearance improves as expected.
Zero is offered as well when the data is positive, since a natural bound at zero is common and is exactly the case a purely proportional ladder would miss.
A profiled bound is chosen for the quality of the fit and is not an estimate of where the support truly ends. The bound is weakly identified: on exponential samples whose true lower bound is zero, profiling has selected bounds far below zero and still fitted well, because the coefficients absorb the difference. Supply the bound directly whenever its value is known.
Parameters
the offsets to try, as fractions of the observed range
whether to offer zero as a lower bound candidate for positive data