Improving confidence intervals and central value estimation in small datasets through hybrid parametric bootstrapping

We developed a hybrid parametric bootstrapping (HPB) method for analyzing small datasets with high precision. This method addresses the challenge of estimating confidence intervals (CI) and central values when traditional distribution assumptions do not apply. Our HPB is combined with Steiner's Most Frequent Value (MFV) technique. The MFV method minimizes the information loss associated with small datasets, while the HPB considers the uncertainty of each separate element. As a practical example, we applied this innovative and robust statistical methodology to refine prior measurements of the half-life of ${}^{97}\text{Ru}$. Using the MFV technique integrated with the HPB method, we obtained a significantly more precise half-life estimate, $T_{1/2,\text{MFV(HPB)}}={2.8385}_{-0.0075}^{+0.0022}$ days. This refined value features a 68.27% confidence interval from 2.8310 to 2.8407 days and a 95.45% confidence interval from 2.8036 to 2.8485 days, as calculated using the percentile method. Our analysis demonstrates a substantial reduction in uncertainty–over 30 times lower than that reported in nuclear data sheets–indicating the potential for widespread analytical impact. In addition, employing alternative minimization strategies can reduce the statistical uncertainty by a further 44%. The HPB method effectively addresses the uncertainties inherent in small datasets, as demonstrated by re-evaluating the specific activity measurements for ${}^{39}\text{Ar}$ using underground data. We report $S{A}_{\text{MFV(HPB)}}={0.966}_{-0.020}^{+0.027}$ Bq/kg_atmAr, with confidence intervals (68.27%: 0.946–0.993; 95.45%: 0.921–1.029) derived using the percentile method. Advances in statistical methods are important for making data analysis more accurate and reliable, especially when combining and interpreting information from different sources. The developed tools help handle complex data more effectively, thereby improving the process and understanding of information in real-world applications where precision is essential.