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We consider the fundamental problem of matching a template to a signal. We doso by M-estimation, which encompasses procedures that are robust to grosserrors (i.e., outliers). Using standard results from empirical process theory,we derive the convergence rate and the asymptotic distribution of theM-estimator under relatively mild assumptions. We also discuss the optimalityof the estimator, both in finite samples in the minimax sense and in thelarge-sample limit in terms of local minimaxity and relative efficiency.Although most of the paper is dedicated to the study of the basic shift modelin the context of a random design, we consider many extensions towards the endof the paper, including more flexible templates, fixed designs, the agnosticsetting, and more.

Template Matching and Change Point Detection by M-estimation