Looking for the same needle in multiple haystacks: Performance bounds

We consider the problem of finding the same pattern in multiple sets. This problem can be applied in a variety of signal processing and machine learning problems including DNA sequencing and detection of electrical signatures. In our problem setting, each set contains only a single unknown pattern of interest among many other patterns. To understand the performance limitations associated with this setting, we focus on the evaluation of the Cramér-Rao lower bound (CRLB). We introduce a probabilistic model for the problem. The random position of a pattern in a given set gives rise to a mixture model and consequently a non trivial CRLB analysis. We present the derivation of the CRLB for the problem and provide a numerical evaluation of the CRLB. We verify our expression for the CRLB against the mean-squared-error of an iterative implementation of the maximum likelihood estimator.