The empty set and zero likelihood problems in maximum empirical likelihood estimation

We describe a previously unnoted problem which, if it occurs, causes the empirical likelihood method to break down. It is related to the empty set problem, recently described in detail by Grendár and Judge (2009), which is the problem that the empirical likelihood model is empty, so that maximum empirical likelihood estimates do not exist. An example is the model that the mean is zero, while all observations are positive. A related problem, which appears to have gone unnoted so far, is what we call the zero likelihood problem. This occurs when the empirical likelihood model is nonempty but all its elements have zero empirical likelihood. Hence, also in this case inference regarding the model under investigation breaks down. An example is the model that the covariance is zero, and the sample consists of monotonically associated observations. In this paper, we define the problem generally and give examples. Although the problem can occur in many situations, we found it to be especially prevalent in marginal modeling of categorical data, when the problem often occurs with probability close to one for large, sparse contingency tables.