Accuracy of the conservative delay-to-distance estimator under exponential noise

Derive an explicit accuracy characterization for the conservative estimator c_hat = min_i(t_i/d_i) used in the trustless Proof of Internet Geometry (PoIG) protocol under a delay model with exponential noise, where t_i/d_i combines an exponential noise term in the numerator with a uniformly sampled distance term in the denominator. Specifically, determine the distribution and/or provide tight bounds on the variance (or an equivalent accuracy measure) of c_hat in this exponential-noise setting to complete the analytical treatment of PoIG accuracy beyond the Gaussian-noise case.

Background

The paper analyzes the accuracy of delay-to-distance mappings in the Proof of Internet Geometry (PoIG) phase. Under a Gaussian-noise model, the authors derive an accuracy expression for the estimator via linear regression. However, when modeling measurement noise as exponential—motivated by the one-sided nature of RTT noise in conservative mappings—the maximum likelihood estimator becomes c_hat = min(t_i/d_i).

Because t_i/d_i mixes exponential noise in the numerator with a random (uniformly sampled) denominator, the authors state they cannot provide a direct analytical accuracy measure. A formal derivation of the estimator’s distribution or variance (or tight bounds) would close this gap and solidify the theoretical guarantees of the trustless PoIG protocol.