ε- SOLUTIONS OF MINIMIZATION PROBLEMS IN STATISTICS

There are various statistical procedures in which one is interested in minimizing a lower semicontinuous function on some complete metric space. For instance in decision theory, likelihood estimation, and minimum distance estimation one minimizes some lower semicontinuous objective function in order to construct an appropriate estimator. There are instances where one cannot implement such procedures because the minimum does not exist. To this end one may consider a regularized problem for which a solution exists. We will present a general result which gives conditions under which one solve the regularized problem.