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Solid-state nanopores have been proposed for rapid and inexpensive deoxyribonucleic acid (DNA) sequencing and analysis. This technology is primarily based on characterizing the ionic current flowing through the pore as DNA translocates from one side of the pore to the other side under the influence of an electric field. The magnitude of the DNA-induced current blockade is an important analytical feature for these applications. However, it remains a challenging task to accurately determine the ionic current levels due to small signal-to-noise ratios. In order to facilitate reliable analysis it is necessary to understand the noise statistics and develop effective signal estimation techniques. In this paper, we conduct a molecular dynamics simulation of DNA translocations through a solid-state nanopore and reveal that the simulated ionic current signals contain both thermal and shot noise. We then develop a model for these signals and propose a maximum likelihood estimator (MLE) for estimating the ionic current levels. We show that the MLE has the potential to significantly outperform the classic sample mean estimator.