Minimax Bahadur Efficiency for Small Confidence Levels

We investigate the problem of estimating an unknown regression function at a fixed point. As the efficiency criterion, we use the risk function initially suggested by R. Bahadur. We construct efficient estimates for the classes of Lipschitz and Hцlder regression functions. The principle of constructing efficient estimates is illustrated by the estimation of the shift parameter, which is a classic example of the parametric problem.