We study the estimation of Zipfian distributions under $L_1$ loss, and provide near minimax optimal bounds in several regimes. Specifically, we assume observations arrive from a known alphabet, and with a known decay rate parametrizing the Zipfian, but we do not know a priori which alphabet elements have larger probability than others. We present a novel Sort and Snap estimator, which uses the empirical proportions to sort the alphabet, and then snaps them to the associated term from the Zipfian distribution. For arbitrary decay rates and smaller alphabet sizes, as well as for large decay rates and large alphabet sizes, we show an exact or minor variant of this estimator is near minimax optimal and has exponential improvement over the standard empirical proportion estimator. However, for small decay rates and larger alphabet sizes a simulation study indicates the standard empirical proportion estimator is competitive with Sort and Snap procedures. In addition to providing nearly tight bounds for important high-dimensional estimation problems, we believe the Sort and Snap estimator, and its analysis, will have independent interest.

MOTIVATION: Many recent advancements in single-molecule localization microscopy exploit the stochastic photoswitching of fluorophores to reveal complex cellular structures beyond the classical diffraction limit. However, this same stochasticity makes counting the number of molecules to high precision extremely challenging, preventing key insight into the cellular structures and processes under observation. RESULTS: Modelling the photoswitching behaviour of a fluorophore as an unobserved continuous time Markov process transitioning between a single fluorescent and multiple dark states, and fully mitigating for missed blinks and false positives, we present a method for computing the exact probability distribution for the number of observed localizations from a single photoswitching fluorophore. This is then extended to provide the probability distribution for the number of localizations in a direct stochastic optical reconstruction microscopy experiment involving an arbitrary number of molecules. We demonstrate that when training data are available to estimate photoswitching rates, the unknown number of molecules can be accurately recovered from the posterior mode of the number of molecules given the number of localizations. Finally, we demonstrate the method on experimental data by quantifying the number of adapter protein linker for activation of T cells on the cell surface of the T-cell immunological synapse. AVAILABILITY AND IMPLEMENTATION: Software and data available at https://github.com/lp1611/mol_count_dstorm. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.