Breaking, and it corrects a wrong answer.
al_scale() now matches the sample band arc length to the
spacing-corrected model band length al_band_model_star()
rather than to the population band length al_band_model().
The old behaviour was inconsistent: order statistics inside the band are
spaced like Q’(u)E/n with E standard exponential, so the polygonal sum
converges to
S*(sigma) = int E sqrt(f0(z)^2 + sigma^2 E^2) dz = E S(sigma E),and not to S(sigma). The two differ by a constant factor, because the
square root is nonlinear and E[E^2] = 2 rather than 1, and the gap does
NOT close as the sample grows. At the standard normal on the default
band the old estimator returned about 1.05 when the scale was 1, at
every sample size. al_scale() now returns 1.001 at n =
20000 and its bias shrinks with n.
Estimates of SCALE change. Regression slopes computed from this estimator do not, because a multiplicative bias in the level cancels in a log-linear slope; interval coverage does change, and becomes correct.
New al_band_model_star(), the spacing-corrected
model band arc length, on both fronts. Its exponential expectation uses
an 80-node Gauss-Legendre rule under the map t = s/(1 - s), which agrees
with adaptive double integration to 3e-13 across sigma from 0.1 to 10
and across four bands. A Gauss-Laguerre rule was tried first and
rejected: it reaches only 2e-5 at sigma = 10, because the integrand has
a kink at t of order f/sigma that falls below its first node.
New al_scale_raw(), the uncorrected estimator,
retained only so that the size of the correction can be measured. It is
inconsistent and is documented as such.
Breaking. k4_fit_varpro() now
searches the fourth shape parameter h DIRECTLY rather than
on the log scale, and the sentinel asking for the free four-parameter
fit changes from a negative hfix to NA
(NaN in C and in Python). Column four of
starts now carries h itself, not
log h.
This is not a refactor. On the log scale h was
necessarily positive and the boundary at h = 0 lay at minus
infinity, so the free fit could not reach the negative half-line at all
and crept toward the boundary reporting a spurious small h
instead of naming the submodel it had effectively chosen. The negative
half-line holds real members – the log-logistic at h = -1,
the Burr III and five-parameter logistic slice at h = -1/m
– and an audit of the edible-oil traces found curves that genuinely
prefer it. h = 0, the generalised extreme value member, is
now an interior point of the free search.
It fails silently if you do not act. Code that
passed hfix = -1 to mean “fit h freely” now gets
h HELD at -1, which fits the log-logistic and returns a
plausible number. Replace any negative hfix meant as a
free-fit request with NA_real_ in R, or
hfix=None in Python. Passing hfix = 0 for the
generalised extreme value submodel is unaffected, as is
NA.
Both fronts move together: the shared C back end, the R wrapper and the Python wrapper all take the new sentinel, and the Python front had been sending -1.0 internally for a free fit.
The quantile-family routines arck4_q(),
arck4_tau34(), arck4_fit_lmom(),
arceq2_bc_q() and arceq2_bc_pdf() are now
registered as C callables and declared in
inst/include/arcstatAPI.h, so a dependent package can call
them from its own C rather than through R. The registration is additive:
nothing already exported changes, no numerics are affected, and dynamic
symbol lookup stays disabled, which is why each entry point is
registered explicitly.
arck4_readings() now requires the density’s grid
maximum to be INTERIOR before reporting a mode. For
k >= 1 with h < 1 both terms of
(log q)'(u) = (h-1)/u - (k-1) h u^(h-1)/(1-u^h) are
negative, so the quantile density is strictly decreasing, the density is
strictly increasing, and the argmax lands on the support endpoint. The
routine previously returned that endpoint as a mode together with two
readings computed from it – finite, plausible-looking, and meaningless.
Both readings are now NaN there, which is correct: neither
is defined when the curve has no interior steepest point. Found while
re-deriving the equivalence locus of the four-parameter kappa family,
where the artefact had been read as a second branch of the
locus.