  It is a standard criterium in statistics to define an optimal estimator the
one with the minimum variance. Thus, the optimality is proved with inequality
among variances of competing estimators. The inequalities, demonstrated here,
disfavor the standard least squares estimators. Inequalities among estimators
are connected to names of Cramer, Rao and Frechet. The standard demonstrations
of these inequalities require very special analytical properties of the
probability functions, globally indicated as regular models. These limiting
conditions are too restrictive to handle realistic problems in track fitting. A
previous extension to heteroscedastic models of the Cramer-Rao-Frechet
inequalities was performed with Gaussian distributions. These demonstrations
proved beyond any possible doubts the superiority of the heteroscedastic models
compared to the standard least squares method. However, the Gaussian
distributions are typical members of the required regular models. Instead, the
realistic probability distributions, encountered in tracker detectors, are very
different from Gaussian distributions. Therefore, to have well grounded set of
inequalities, the limitations to regular models must be overtaken. The aim of
this paper is to demonstrate the inequalities for least squares estimators for
irregular models of probabilities, explicitly excluded by the
Cramer-Rao-Frechet demonstrations. Estimators for straight and parabolic tracks
will be considered. The final part deals with the form of the distributions of
simplified heteroscedastic track models reconstructed with optimal estimators
and the standard (non-optimal) estimators. A comparison among the distributions
of these different estimators shows the large loss in resolution of the
standard least-squares estimators.
