On Some Singularities in Problems of Parameter Estimation

Assume that there are two identical remote objects and we want to estimate the distance to the closest of them and the distance between objects. For that purpose, some device (like a radar) is used, and the observed signal is the sum of signals reflected by each object. Moreover, the observed signal is corrupted by white Gaussian noise. Singularity (i.e., very poor estimation accuracy) occurs in this problem if objects are very close to each other. Our aim is to demonstrate this inevitable singularity by the example of the maximum likelihood estimate and also to show that it takes place for any other estimate.