We estimate on a compact interval densities with isolated irregularities, such as discontinuities or discontinuities in some derivatives. From independent and identically distributed observations we construct a kernel estimator with non-constant bandwidth, in particular in the vicinity of irregularities. It attains faster rates, for the risk $L_{p}, p\geq 1$, than usual estimators with a fixed global bandwidth. Optimality of the rate found is established by a lower bound result. We then propose an adaptive method inspired by Lepski's method for automatically selecting the variable bandwidth, without any knowledge of the regularity of the density nor of the points where the regularity breaks down. The procedure is illustrated numerically on examples.

翻译：我们估计在紧区间上具有孤立不规则性（如间断点或某些导数间断）的密度函数。基于独立同分布观测数据，我们构建了一种具有非恒定带宽的核估计器，特别针对不规则点邻域进行优化。在$L_{p}, p\geq 1$风险度量下，该估计器相比采用固定全局带宽的传统估计器能达到更快的收敛速率。通过下界结果证明了所得速率的最优性。随后，我们提出一种受Lepski方法启发的自适应方法，可在无需已知密度正则性信息及正则性突变点位置的情况下，自动选择可变带宽。通过数值算例对该方法的有效性进行了验证。