On the available local asymptotic efficiency of some goodness-of-fit criteria

The concept of local asymptotic Bahadur optimality for a sequence of statistics is well known. The following problem was posed and in many cases solved by Ya.Yu. Nikitin: to describe the class of distributions for which a given sequence of statistics is locally asymptotically optimal against specific alternatives (shift, scale, etc.). However, in some cases this problem has no solution. We propose to consider a modified problem. Its solution gives the value of the available asymptotic efficiency of the corresponding statistics.