Quasi-ML hop period estimation from incomplete data

Given a noisy sequence of (possibly shifted) integer multiples of a certain period, it is often of interest to estimate the period (and offset). With known integer regressors, the problem is classical linear regression. In many applications, however, the actual regressors are unknown; only categorical information (i.e., the regressors are integers) and, perhaps, loose bounds are available. Examples include hop timing estimation, pulse repetition interval (PRI) analysis, and passive rotating-beam radio scanning. With unknown regressors, this seemingly simple problem exhibits many surprising twists. Even for small sample sizes, a proposed quasi-maximum likelihood approach essentially meets the clairvoyant CRB at moderately high SNR - the latter assumes knowledge of the unknown regressors. This is quite unusual, and it holds despite the fact that our algorithm ignores noise color. We outline analogies and differences between our problem and classical linear regression and harmonic retrieval, and corroborate our findings with careful simulations.