UNIFORM CONVERGENCE RATES FOR NONPARAMETRIC ESTIMATORS OF A DENSITY FUNCTION AND ITS DERIVATIVES WHEN THE DENSITY HAS A KNOWN POLE
研究了密度函数存在已知极点时，其非参数估计量及其导数的均匀收敛速度，这对拍卖、劳动和消费者搜索等微观计量模型中的半参数估计很重要。
We study the uniform convergence rates of nonparametric estimators for a probability density function and its derivatives when the density has a known pole. Such situations arise in some structural microeconometric models, for example, in auction, labor, and consumer search, where uniform convergence rates of density functions are important for nonparametric and semiparametric estimation. Existing uniform convergence rates based on Rosenblatt’s kernel estimator are derived under the assumption that the density is bounded. They are not applicable when there is a pole in the density. We treat the pole nonparametrically and show various kernel-based estimators can attain any convergence rate that is slower than the optimal rate when the density is bounded uniformly over an appropriately expanding support under mild conditions.