Unbiased estimators are derived for a sample from the geometric density with unknown $p$ and unknown location parameter. Mean square errors are compared with the maximum likelihood estimator and unbiased tests of hypotheses are given.
"The Geometric Density with Unknown Location Parameter." Ann. Math. Statist. 41 (3) 1078 - 1082, June, 1970. https://doi.org/10.1214/aoms/1177696986