Minimax principle is a very useful concept in mathematical statistics for finding optimal estimators. While unbiasedness and invariance principle are useful tools for finding optimal estimators, they are often restrictive and in certain cases may not even yield optimal estimators see Ferguson (1967). Minimax principle on the other hand, is based on linear ordering principle and is often less restrictive. While there are several methods for finding minimax optimal estimators such as methods due to H\\acute{a}jek, Le Cam, Fano and Assouad, in our work, we specifically use H\\acute{a}jek and Fano's methods to explore the minimax optimality of integral curve estimators in high order tensor models. High angular resonance diffusion imaging (HARDI) is a popular in-vivo brain imaging technique proposed by \\ddot{O}zarslan and Mareci (2003). Besides the mathematical model for HARDI, successful tracing of neural fibers using HARDI presents the challenge of estimation and uncertainty quantification in presence of measurement errors. Our work here is based on the semi-parametric estimation method proposed by Carmichael and Sakhanenko (2015), where the authors have provided a consistent method for tracing fiber in the presence of measurement error using HARDI. The first work described here establishes the estimators proposed in Carmichael and Sakhanenko (2015) are minimax optimal with respect to their asymptotic risk. The framework of HARDI allows to accommodate complex neural fiber structures where fiber tracts cross each other, converge, diverge, "fan out" or "kiss", thus our work generalizes the minimax lower bound results in Sakhanenko (2012) where a similar result was established under a simpler model where imaging signals are modeled by a vector field perturbed by an additive noise. The second work establishes the global bounds for the integral curve estimators proposed by Carmichael and Sakhanenko (2015). Therefore, suggesting that not only the asymptotic rate of convergence of the integral curve estimator is minimax optimal locally but also it is minimax optimal globally. Additionally, in the simulation study of our second work we have introduced a metric based on global minimax optimal rates which can compare the relative accuracy of different imaging protocols that are used to obtain HARDI data.
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