The Cramer–Rao–Frechet inequality is reviewed and extended to track fitting. A diffused opinion attributes to this inequality the limitation of the resolution of the track fits with the number N of observations. It will be shown that this opinion is incorrect, the weighted least squares method is not subjected to that N-limitation and the resolution can be improved beyond those limits. In previous publications, simulations with realistic models and simple Gaussian models produced interesting results: linear growths of the peaks of the distributions of the fitted parameters with the number N of observations, much faster than the $\sqrt{\mathrm{N}}$ of the standard least-squares. These results could be considered a violation of a well-known $1/\mathrm{N}$ -rule for the variance of an unbiased estimator, frequently reported as the Cramer–Rao–Frechet bound. To clarify this point beyond any doubt, a direct proof of the consistency of those results with this inequality would be essential. Unfortunately, such proof is lacking. Hence, the Cramer–Rao–Frechet developments are applied to prove the efficiency (optimality) of the simple Gaussian model and the consistency of the linear growth. The inequality remains valid even for irregular models supporting the similar improvement of resolution for the realistic models.

A standard criterium in statistics is to define an optimal estimator as the one with the minimum variance. Thus, the optimality is proved with inequality among variances of competing estimators. The demonstrations of inequalities among estimators are essentially based on the Cramer, Rao and Frechet methods. They require special analytical properties of the probability functions, globally indicated as regular models. With an extension of the Cramer–Rao–Frechet inequalities and Gaussian distributions, it was proved the optimality (efficiency) of the heteroscedastic estimators compared to any other linear estimator. However, the Gaussian distributions are a too restrictive selection to cover all the realistic properties of track fitting. Therefore, a well-grounded set of inequalities must overtake the limitations to regular models. Hence, the inequalities for least-squares estimators are generalized to any model of probabilities. The new inequalities confirm the results obtained for the Gaussian distributions and generalize them to any irregular or regular model. Estimators for straight and curved tracks are considered. The second part deals with the shapes 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.

Probability distributions for the center of gravity are fundamental tools for track fitting. The center of gravity is a widespread algorithm for position reconstruction in tracker detectors for particle physics. Its standard use is always accompanied by an easy guess (Gaussian) for the probability distribution of the positioning errors. This incorrect assumption degrades the results of the fit. The explicit error forms evident Cauchy–Agnesi tails that render the use of variance minimizations problematic. Therefore, it is important to report probability distributions for some combinations of random variables essential for track fitting: $x=\xi /(\xi +\mu )$, $y=(\xi -\mu )/\left[2(\xi +\mu )\right]$, $w=\xi /\mu$, $x=\theta (x_3-x_1)(-x_3)/(x_3+x_2)+\theta (x_1-x_3)x_1/(x_1+x_2)$ and $x=(x_1-x_3)/(x_1+x_2+x_3)$. The first two are partial forms of the two strip center of gravity. The fourth is the complete two strip center of gravity, and the fifth is a partial form of the three strip center of gravity. For the complexity of the forth equation, only approximate expressions of the probability are allowed. Analytical expressions are calculated assuming $\xi$, $\mu$, $x_1$, $x_2$ and $x_3$ independent Gaussian random variables. The analytical form of the probability for the two strip center of gravity allows one to construct an approximate proof for the lucky model of our previous paper. This proof also suggests how to complete the lucky model by its absence of a scaling constant, relevant to combine different detector types. This advanced lucky model (the super-lucky model) can be directly used in trackers composed of non-identical detectors. The construction of the super-lucky model is very simple. Simulations with this upgraded tool also show resolution improvements for a combination of two types of very different detectors, near to the resolutions of the schematic model.

The construction of a well tuned probability distributions is illustrated in synthetic way, these probability distributions produce faithful realizations of the impact point distributions for particles in silicon strip detector. Their use for track fitting shows a drastic improvements of a factor two, for the low noise case, and a factor three, for the high noise case, respect to the standard approach. The tracks are well reconstructed even in presence of hits with large errors, with a surprising effect of hit discarding. The applications illustrated are simulations of the PAMELA tracker, but other type of trackers can be handled similarly. The probability distributions are calculated for the center of gravity algorithms, and they are very different from gaussian probabilities. These differences are crucial to accurately reconstruct tracks with high error hits and to produce the effective discarding of the too noisy hits (outliers). The similarity of our distributions with the Cauchy distribution forced us to abandon the standard deviation for our comparisons and instead use the full width at half maximum. A set of mathematical approaches must be developed for these applications, some of them are standard in wide sense, even if very complex. One is essential and, in its absence, all the others are useless. Therefore, in this paper, we report the details of this critical approach. It extracts physical properties of the detectors, and allows the insertion of the functional dependence from the impact point in the probability distributions. Other papers will be dedicated to the remaining parts.

An experimental set up, dedicated to isolate an error present in the $\eta$-algorithm, gave an unexpected result. The average of a center of gravity algorithm at orthogonal particle incidence turns out to be non zero. This non zero average signals an asymmetry in the response function of the strips, and introduces a further parameter in the corrections: the shift of the strip response center of gravity respect its geometrical position. A strategy to extract this parameter from a standard data set is discussed. Some simulations with various asymmetric response functions are explored for this test. The method is able to detect easily the asymmetry parameters introduced in the simulations. Its robustness is tested against angular rotations, and we see an almost linear variation with the angle. This simple property is used to simulate a determination of a Lorentz angle with and without the asymmetry of the response function.