The content explores a family of centrality estimators for probability density function (PDF) fitting. The key highlights are:
The authors introduce the Hölder and Lehmer centrality measures as alternatives to the maximum likelihood estimator (MLE). These centrality measures allow for more robust and accurate PDF fitting by incorporating data selection criteria and overcoming the IID assumption.
The Hölder centrality (H-C) is defined as the weighted arithmetic mean of transformed PDF values, where the transformation is controlled by a parameter α. The Lehmer centrality (L-C) is defined as the ratio of weighted sums of transformed PDF values.
The authors establish properties of the H-C and L-C, such as their relationship to the geometric and arithmetic means, their monotonicity in α, and their interpretation as probabilities of observations falling within a cell.
The authors derive the critical points and maximum points of the H-C and L-C, showing that they are related but not necessarily equivalent to the MLE.
The authors propose two measures to evaluate the accuracy of the centrality estimators: the residual error and the observed centrality-Fisher information. These measures provide insights into the uncertainty and shape of the fitted PDF.
A case study is presented for the exponential PDF, demonstrating the application of the centrality estimators and the analysis of their properties.
Overall, the content introduces a generalized framework for robust and accurate PDF fitting using centrality estimators, providing a new perspective on the limitations of the MLE and offering alternative approaches with desirable properties.
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by Djemel Ziou at arxiv.org 04-10-2024
https://arxiv.org/pdf/2404.05816.pdfDeeper Inquiries