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Optical beam center position on an array of detectors is an important parameter that is essential for estimating the angle-of-arrival of the incoming signal beam. In this paper, we have examined the beam position estimation problem for photon-counting detector arrays, and to this end, we have derived and analyzed the <italic>Cram&#x00E9;r-Rao lower bounds</italic> on the mean-square error of unbiased estimators of beam position. Furthermore, we have also derived the Cram&#x00E9;r-Rao lower bounds of other system parameters such as signal peak intensity, and dark current noise power, on the array. In this sense, we have considered robust estimation of beam position in which none of the parameters are assumed to be known beforehand. Additionally, we have derived the Cram&#x00E9;r-Rao lower bounds of beam and noise parameters for observations based on both pilot and data symbols of a <italic>pulse position modulation</italic> (PPM) scheme. Finally, we have considered a two-step estimation problem in which the signal peak and dark current noise intensities are estimated using a <italic>method of moments</italic> estimator, and the beam center position is estimated with the help of a <italic>maximum likelihood</italic> estimator.