Statistical analysis of self-similar conservative fragmentation chains

TL;DR

This paper investigates statistical inference methods for self-similar conservative fragmentation chains, focusing on estimation rates and challenges when only approximate fragment size data is available, with applications in mineral crushing.

Contribution

It provides new estimation rate results for both parametric and nonparametric cases in the context of fragmentation chains, connecting the problem to ill-posed inverse problems.

Findings

Derived upper and lower estimation rates in a parametric framework.

Showed nonparametric estimation difficulty parallels ill-posed inverse problems.

Applicable to mineral crushing and similar fragmentation processes.

Abstract

We explore statistical inference in self-similar conservative fragmentation chains when only approximate observations of the sizes of the fragments below a given threshold are available. This framework, introduced by Bertoin and Martinez [Adv. Appl. Probab. 37 (2005) 553--570], is motivated by mineral crushing in the mining industry. The underlying object that can be identified from the data is the step distribution of the random walk associated with a randomly tagged fragment that evolves along the genealogical tree representation of the fragmentation process. We compute upper and lower rates of estimation in a parametric framework and show that in the nonparametric case, the difficulty of the estimation is comparable to ill-posed linear inverse problems of order 1 in signal denoising.