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Beyond the $\sqrt{N}$ -Limit of the Least Squares Resolution and the Lucky Model

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## Abstract

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## 1. Introduction

## 2. A Simple Gaussian Model and the Linear Growth

`randn`function). Each one is multiplied by one of the two standard deviations ($0.18$, $0.018$) with the given relative probability ($80\%$, $20\%$). The data are scrambled and recollected to simulate a set of parallel tracks crossing few detector layers. This data collection simulates a set of tracks populating a large portion of the tracker system with slightly non-parallel strips on different layers (as it always is in real detectors). The detector layers are supposed parallel, but for the non-parallelism of the strips, the hit positions of a track, relative to the strip centers, seem to be uncorrelated. The properties of the binomial distribution produce the linear growth. Due to the translation invariance, the tracks can be expressed by a single equation ${x}_{i}=\beta +\gamma {y}_{i}$ with $\beta =0$ and $\gamma =0$ for the orthogonal incidence of the set of tracks ($\beta $ is the impact point of the track, $\gamma $ the direction, ${y}_{i}$ the positions of the detector layers and ${x}_{i}$ the hit position). The distributions of ${\gamma}_{f}$, the $\gamma $ value given by the fit, are the object of our study. The $\gamma $ distribution is a Dirac $\delta $-function, as it is usual to test the resolution of the fitted values ${\gamma}_{f}$. The distance of first and the last detector layer is the length of the PAMELA [16] tracker (445 $\mathrm{mm}$, 7063.5 in strip width, for different lengths ${\gamma}_{f}$ scales as usual in least-squares). Other “detector layers” are inserted symmetrically in this length. Two different least squares fits are compared. One suppose identical $\sigma $ of each hit and utilizes the usual equations for homoscedastic systems (we call this standard fit). The other fit applies the appropriate $\sigma $-values to each hit. This second fit shows a linear growth in the resolution (as far as a set of random variables can follow this rule). We generate 150,000 tracks for each configuration.

## 3. The Schematic Model

## 4. The Lucky Model

#### The Linear Growth in the Lucky Model

## 5. Hints for an Experimental Verification

## 6. Linear Growth of the Maximum Likelihood Evaluations

#### Beyond the Present Maximum Likelihood Evaluations

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

probability density function | |

MIP | minimum ionizing particle |

COG | center of gravity |

COG${}_{n}$ | center of gravity algorithm with n-strips |

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**Figure 1.**The red lines are the values of $\sigma $ for the Gaussian model. The blue dots are the effective $\sigma $ of the schematic model. The dimensions are in strip width ($63\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$). The $\eta $-algorithm gives the hit positions, with zero as the strip center.

**Figure 2.**The Gaussian model. The blue lines are the ${\gamma}_{f}$-distributions of the standard fit. The red lines are the ${\gamma}_{f}$-distributions of the $\sigma $-weighted least squares, they show a linear growth in the resolution. The first distribution is centered on zero, the others are shifted by $N-2$ identical steps.

**Figure 3.**The Gaussian model. Extraction of an approximate linear growth from the standard least squares with the selection of low ${\psi}^{2}$-values.

**Figure 4.**Schematic model. The blue ${\gamma}_{f}$ distributions are the standard least squares fits. The red ${\gamma}_{f}$ distributions show the linear growth and are produced with the effective $\sigma $ for each hit.

**Figure 5.**Scatter plot of the effective $\sigma $ as function of the COG${}_{2}$ (${x}_{g2}$) scaled to reach an approximate overlap with the COG${}_{2}$ histogram. More precisely a merged set of red dots corresponding to the blue dots. They are obtained as by-product of the $\eta $-algorithm.

**Figure 6.**The lucky model. The blue distributions are the standard least squares. The red distributions are the ${\gamma}_{f}$ given by the lucky model, very near to those of the schematic model.

**Figure 7.**The maximum likelihood evaluations. The blue distributions are the standard least squares. The red distributions are the ${\gamma}_{f}$ given by the schematic model, the black distributions are the maximum likelihood evaluations, their empirical PDFs are partially covered by those of the schematic model, however their higher maxima are clearly visible.

**Figure 8.**The asterisks are the ${H}_{G}=1/\left(\sqrt{2\pi \phantom{\rule{0.166667em}{0ex}}{\mathrm{\Sigma}}_{D}}\right)$ the heights of a Gaussian of $\sqrt{{\mathrm{\Sigma}}_{D}}$ as standard deviation. The circles are the maxima of the ${\gamma}_{f}$ empirical PDFs. The color code is that of Figure 7.

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Landi, G.; Landi, G.E. Beyond the *Instruments* **2022**, *6*, 10.
https://doi.org/10.3390/instruments6010010

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Landi G, Landi GE. Beyond the *Instruments*. 2022; 6(1):10.
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Landi, Gregorio, and Giovanni E. Landi. 2022. "Beyond the *Instruments* 6, no. 1: 10.
https://doi.org/10.3390/instruments6010010