Nick Cox wrote about fitting distributions with -nl-. You can use this
approach with -nl- (and -amoeba-) for fitting Johnson distributions, too,
following Swain, Venkatraman and Wilson (1988). When you're in a region of
the parameter space that is numerically nasty, such as is not infrequently
the case with Johnson SB distributions, fitting by least squares like this
often fails in my experience, even given starting values that are in the
neighborhood of the minimum (maximum). Failure here means either failure to
converge after a reasonable number of iterations, or wandering off and
converging at a local minimum where the estimates are outside the parameter
space. Sometimes in these cases, even when fed starting values right at the
minimum, although it might not wander off, -nl- will give missing values for
the standard errors of the coefficients, and associated t statistics and
p-values for one or more of the Johnson parameter estimates. It seems
that the curvature is nearly nil (or nearly infinite) at the minimum in
these nasty cases (see Karian and Dudewicz). I've heard that least absolute
deviation works better than least squares in these circumstances, but
haven't
looked into it. Putting constraints on parameter estimates in -nl- could
also help in using it to fit Johnson distributions.
Joseph Coveney
J. J. Swain, S. Venkatraman and J. R. Wilson, Least-squares estimation of
distribution functions in Johnson's translation system. J Statist Comput
Simul 29:271--87, 1988.
Z. A. Karian and E. J. Dudewicz, Computational issues in fitting statistical
distributions to data.
www.stat.auburn.edu/fsdd2006/papers/KarianDudewiczmain.pdf
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