Jittering estimators are nonparametric function estimators for mixed data. They extend arbitrary estimators from the continuous setting by adding random noise to discrete variables. We give an in-depth analysis of the jittering kernel density estimator, which reveals several appealing properties. The estimator is strongly consistent, asymptotically normal, and unbiased for discrete variables. It converges at minimax-optimal rates, which are established as a by-product of our analysis. To understand the effect of adding noise, we further study its asymptotic efficiency and finite sample bias in the univariate discrete case. Simulations show that the estimator is competitive on finite samples. The analysis suggests that similar properties can be expected for other jittering estimators.

中文翻译：

---

抖动核密度估计器的渐近分析

抖动估计器是混合数据的非参数函数估计器。通过向离散变量添加随机噪声，它们从连续设置扩展了任意估计量。我们对抖动内核密度估计器进行了深入分析，揭示了几个吸引人的特性。估计量是强一致的，渐近正态的，对于离散变量没有偏见。它以最小最大最优速率收敛，这是我们分析的副产品。为了了解添加噪声的影响，我们进一步研究了单变量离散情况下的渐近效率和有限样本偏差。仿真表明，估计器在有限样本上具有竞争力。分析表明，对于其他抖动估计器，可以期望具有类似的特性。