Asymptotic distribution theory for contamination models

In many situations one is interested in identifying observations that come from sources of variation other than the normal background or baseline source. A simple model for such situations is a two point mixture model where one component in the mixture corresponds to the baseline model and the second to the other sources (the contamination component). Here the goal is two-fold: (i) detect the overall presence of Contamination and (ii) identify observations that may be contaminated. A locally most powerful test is presented which gives some insights on how to accomplish this. Surprisingly, the test statistic can have an asymptotic distribution that is based on a stable law that is not the normal distribution. Examples and simulations are given to illustrate the approach