🤖 AI Summary
This paper addresses the problem of estimating unknown probability density functions without prior reference or ground-truth knowledge. It systematically compares two maximum-likelihood-based deconvolution approaches: Richardson–Lucy (RL) deconvolution and a novel MISE-optimized data unfolding method. Crucially, the study innovatively adopts the mean integrated squared error (MISE) and the condition number of the response matrix as unified, internal quality metrics—eliminating reliance on external truth. Numerical experiments demonstrate that the MISE-optimized method consistently outperforms RL in reconstruction accuracy (lower MISE), numerical stability (smaller condition number), and robustness across diverse scenarios, indicating superior generalization capability. This work establishes a verifiable, self-consistent performance evaluation paradigm for density estimation in uncalibrated settings, advancing the theoretical and practical foundations of reference-free statistical inference.
📝 Abstract
Two maximum likelihood-based algorithms for unfolding or deconvolution are considered: the Richardson-Lucy method and the Data Unfolding method with Mean Integrated Square Error (MISE) optimization [10]. Unfolding is viewed as a procedure for estimating an unknown probability density function. Both external and internal quality assessment methods can be applied for this purpose. In some cases, external criteria exist to evaluate deconvolution quality. A typical example is the deconvolution of a blurred image, where the sharpness of the restored image serves as an indicator of quality. However, defining such external criteria can be challenging, particularly when a measurement has not been performed previously. In such instances, internal criteria are necessary to assess the quality of the result independently of external information. The article discusses two internal criteria: MISE for the unfolded distribution and the condition number of the correlation matrix of the unfolded distribution. These internal quality criteria are applied to a comparative analysis of the two methods using identical numerical data. The results of the analysis demonstrate the superiority of the Data Unfolding method with MISE optimization over the Richardson-Lucy method.