There is an intense and partly recent literature focussing on the problem of selecting the bandwidth parameter for kernel density estimators. Available methods are largely `very nonparametric', in the sense of not requiring any knowledge about the underlying density, or `very parametric', like the normality-based reference rule. This report aims at widening the scope towards the inclusion of many semiparametric bandwidth selectors, via Hermite type expansions aroundthe normal distribution. The resulting bandwidths may be seen as carrying out suitable corrections on the normal reference rule, requiring a low number of extra coefficients to be estimated from data. The present report introduces and discusses some basic ideas and develops the necessary initial theory, but modestly chooses to stop short of giving precise recommendations for specific procedures among the many possible constructions. This will require some further analysis, numerical work, and some simulation-based exploration.

翻译：关于核密度估计中带宽参数选择问题的研究文献丰富且部分成果较为新颖。现有方法大多属于"高度非参数化"类型，即无需任何关于基础分布的先验知识，或如基于正态分布的参考规则等"高度参数化"方法。本报告旨在通过围绕正态分布的埃尔米特型展开，将研究范畴扩展至涵盖多种半参数带宽选择方法。所得带宽可视为对正态参考规则进行适当修正的结果，仅需从数据中估计少量额外系数。本报告提出并探讨了若干基本思路，发展了必要的初步理论，但审慎地暂未在众多可能构建的方案中给出具体程序的精确建议。这需要进一步的理论分析、数值计算及基于模拟的探索研究。