Two new fitting methods are explored for momentum reconstruction. They give a substantial increase of momentum resolution compared to standard fit. The key point is the use of a different (realistic) probability distribution for each hit (heteroscedasticity). In the first fitting method an effective variance is calculated for each hit, the second method uses the search of the maximum likelihood. The tracker model is similar to the PAMELA tracker with its two sided detectors. Here, each side is simulated as a momentum reconstruction device. One of the two is similar to silicon micro-strip detectors of large use in running experiments. The gain obtained in momentum resolution is measured as the virtual magnetic field and the virtual signal-to-noise ratio required by the standard fits to overlap with the best of the new methods. For the low noise side, the virtual magnetic field must be increased 1.5 times to reach the overlap and 1.8 for the other. For the high noise side, the increases must be 1.8 and 2.0. The signal-to-noise ratio has to be increased by 1.6 for the low noise side and 2.2 for the high noise side ( $\eta$ -algorithms). Each one of our two methods shows a very rapid linear increase of the resolution with the number N of detector layers, the two standard fits have the usual slow growth less than $\sqrt{\mathrm{N}}$ .

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.

Silicon micro-strip detectors are fundamental tools for the high energy physics. Each detector is formed by a large set of parallel narrow strips of special surface treatments (diode junctions) on a slab of very high quality silicon crystals. Their development and use required a large amount of work and research. A very synthetic view is given of these important components and of their applications. Some details are devoted to the basic subject of the track reconstruction in silicon strip trackers. Recent demonstrations substantially modified the usual understanding of this argument.

A very simple Gaussian model is used to illustrate an interesting fitting result: a linear growth of the resolution with the number N of detecting layers. This rule is well beyond the well-known rule proportional to $\sqrt{N}$ for the resolution of the usual fits. The effect is obtained with the appropriate form of the variance for each hit (observation). The model reconstructs straight tracks with N parallel detecting layers, the track direction is the selected parameter to test the resolution. The results of the Gaussian model are compared with realistic simulations of silicon micro-strip detectors. These realistic simulations suggest an easy method to select the essential weights for the fit: the lucky model. Preliminary results of the lucky model show an excellent reproduction of the linear growth of the resolution, very similar to that given by realistic simulations. The maximum likelihood evaluations complete this exploration of the growth in resolution.

To complete a previous paper, the probability density functions of the center-of-gravity as positioning algorithm are derived with classical methods. These methods, as suggested by the textbook of Probability, require the preliminary calculation of the cumulative distribution functions. They are more complicated than those previously used for these tasks. In any case, the cumulative probability distributions could be useful. The combinations of random variables are those essential for track fitting $x={\xi}/{(\xi+\eta)}$, $x=\theta(x_3-x_1) (-x_3)/(x_3+x_2) +\theta(x_1-x_3)x_1/(x_1+x_2)$ and $x=(x_1-x_3)/(x_1+x_2+x_3)$. The first combination is a partial form of the two strip center-of-gravity. The second is the complete form, and the third is a simplified form of the three strip center-of-gravity. The cumulative probability distribution of the first expression was reported in the previous publications. The standard assumption is that $\xi$, $\eta$, $x_1$, $x_2$ and $x_3$ are independent random variables.

The Cramer-Rao-Frechet inequality is reviewed specializing it 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 turns out that this opinion is incorrect, weighted least squares method is not subjected to that N-limitation. In a previous publication, simulations with a simple Gaussian model produced interesting results: a linear growth of the peaks of the distributions with the number $N$ of observations, much faster than the $\sqrt{N}$ of the standard least squares. These results could be considered a violation of a well known $1/N$-rule for the variance of an unbiased estimator, frequently reported as the Cramer-Rao-Frechet bound. To clarify this point beyond any doubt, it would be essential a direct proof of the consistency of those results with this inequality. Unfortunately, such proof is lacking or very difficult to find. Hence, the Cramer-Rao-Frechet developments are applied to prove the efficiency (optimality) of the simple Gaussian model and the consistency of its results. The inequality remains valid even for irregular models supporting the results of realistic models with similar growths.