Precision in peak parameter estimation for Gaussian and Lorentzian profiles: Guidelines for instrument optimization

The analysis of signal data plays a fundamental role across diverse scientific fields, where the high-precision estimation of peak parameters such as intensity, position, bandwidth, and area is essential for advancing scientific progress. However, understanding how these parameters are influenced by profile shape and instrumentation remains a key challenge, as these factors fundamentally determine the precision and efficiency of analytical measurements. To address this, we derive analytical solutions for the precision of Lorentzian profile peak parameters under Poisson noise constraints using the Cramér–Rao inequality and Fisher information. By comparing these precision limits with those of Gaussian profiles, we identify the peak characteristics that strongly influence parameter precision. Our results demonstrate that the profile slope and tail intensity are critical factors for estimating peak position and position differences, with Gaussian profiles exhibiting higher precision under identical conditions. Conversely, the weaker intensity–bandwidth correlation in Gaussian profiles contributes to their superior precision in intensity and bandwidth estimation. Additionally, Lorentzian profiles show a <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:msqrt> <a:mn>3</a:mn> </a:msqrt> </a:math> -fold improvement in area ratio precision over intensity ratio precision, surpassing the corresponding improvement in Gaussian profiles. This distinction arises from the larger area and stronger intensity–bandwidth correlation inherent in Lorentzian profiles. These findings highlight the significant impact of profile shape on peak parameter precision, providing essential guidance for decision-making in instrument design and analytical conditions. By integrating these analytical solutions with insights from analytical chemistry, we establish a theoretical framework linking peak parameter precision with instrument performance. This framework enables quantitative evaluation of the effects of equipment and experimental design improvements on measurement precision. As a practical application, we numerically demonstrate how enhanced instrument conditions improve the precision of isotope ratio measurements of <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mi mathvariant="normal">C</b:mi> <b:msub> <b:mi mathvariant="normal">O</b:mi> <b:mn>2</b:mn> </b:msub> </b:mrow> </b:math> using Raman spectroscopy. Our findings provide practical guidelines for instrument optimization and budget planning, ultimately supporting high-precision analytical applications.