Consequences of lack of smoothness in nonparametric estimation (in Russian)

Nonparametric estimation is widely used in statistics and econometrics with many asymptotic results relying on smoothness of the underlying distribution, however, there are cases where such assumptions may not hold in practice. Lack of smoothness may have undesirable consequences such as an incorrect choice of window width, large estimation biases and incorrect inference. Optimal combinations of estimators based on different kernel/bandwidth can achieve automatically the best unknown rate of convergence. The combined estimator was successfully applied in density estimation, estimation of average derivatives and for smoothed maximum score in a binary choice model. In the extreme case when density does not exist the estimator "estimates" a non-existent function; nevertheless its limit process can be described in terms of generalized (in terms of generalized functions) Gaussian processes. Inference about existence of density and about its smoothness is not yet well developed; some preliminary results are discussed.